On Properties of Certain Special Zeros of Functions in the Selberg Class

Abstract

In this paper, assuming generalized Riemann hypothesis, we give an upper bound for the multiplicity of eventual zero at central point 1 / 2 and location of the first zero with positive imaginary part of function in a certain subclass of the extended Selberg class. We apply our results to automorphic L-functions attached to irreducible unitary automorphic representations of $GL_N(\mathbb {Q})$.

1 Introduction

In 1989, Selberg  [15] defined a general class of Dirichlet series having an Euler product, analytic continuation and a functional equation of Riemann type (plus some side conditions), and formulated some fundamental conjectures concerning them. Especially these conjectures give this class of Dirichlet series a certain structure which applies to central problems in number theory.

The Selberg class of functions, denoted by $\mathcal {S}$, is a general class of Dirichlet series F satisfying the following properties:

• (1) (Dirichlet series) F posses a Dirichlet series representation

  $$\begin{aligned} F(s)=\sum _{n=1}^{\infty }\frac{a_F(n)}{n^s}, \end{aligned}$$

  that converges absolutely for $\mathfrak {R}s >1$.

• (2) (Analytic continuation) There exists an integer $m \ge 0$ such that the function $(s-1)^m F(s)$ is an entire function of finite order. The smallest such number is denoted by $m_F$ and is called the polar order of F.

• (3) (Functional equation) The function F satisfies the functional equation

  $$\begin{aligned} {\varPhi }_F (s)= w \overline{{\varPhi }_F(1-\bar{s})}, \end{aligned}$$

  where

  $$\begin{aligned} {\varPhi }_F(s)= F(s) Q_F^s \prod _{j=1}^{r} {\varGamma }(\lambda _j s + \mu _j), \end{aligned}$$

  with $Q_F> 0, r \ge 0, \lambda _j > 0, |w|=1, \mathfrak {R}(\mu _j) \ge 0, j = 1, \ldots , r$.

• (4) (Ramanujan hypothesis) For every $\epsilon > 0$ we have $a_F (n) \ll n^\epsilon$.

• (5) (Euler product)

  $$\begin{aligned} \log F(s)=\sum _{n=1}^{\infty }\frac{b_F(n)}{n^s}, \end{aligned}$$

  where $b_F(n)=0$ for all $n\ne p^m$ with $m \ge 1$ and p prime, and $b_F(n) \ll n^\theta$ for some $\theta < 1/2.$

We also recall that degree and conductor, defined by

$$\begin{aligned} d_F = 2 \sum _{j=1}^{r} \lambda _j, \quad q_F = (2\pi )^{d_F} Q_F^2 \prod _{j=1}^{r} \lambda _j^{2\lambda _j}, \end{aligned}$$

(1)

respectively, are invariants of $F \in \mathcal {S}$ (see  [8]).

In fact, by the conductor hypothesis it is assumed that for every $F \in \mathcal {S}$ one has $q_F \in \mathbb {N}$. In the special case when $F(s)=\zeta (s)$, where $\zeta (s)$ is the Riemann zeta function then $q_{\zeta }=1$. If $F(s)=\zeta _K(s)$, where $\zeta _K(s)$ is the Dedekind zeta function of a number field K, then $q_{\zeta _K}=|d_K|$ (see e.g.  [12]). The extended Selberg class $\mathcal {S}^{\sharp }$, introduced in  [7], is the class of functions satisfying axioms (1), (2) and (3). For more information on properties of Selberg class and extended Selberg class see e.g.  [1], [6, 12] and [13].

It is conjectured that the Selberg class coincides with the class of all automorphic L-functions.

In order to apply some of our results unconditionally to automorphic L-functions attached to irreducible unitary automorphic representations of $GL_N(\mathbb {Q})$, we also consider class $\mathcal {S}^{\sharp \flat }$, introduced in  [10]. It consists of functions satisfying axioms (1), (2) and the two following axioms:

• (3’) (Functional equation) The function F satisfies the functional equation

  $$\begin{aligned} {\varPhi }_F (s)= w \overline{{\varPhi }_F(1-\bar{s})}, \end{aligned}$$

  where

  $$\begin{aligned} {\varPhi }_F(s)= F(s) Q_F^s \prod _{j=1}^{r} {\varGamma }(\lambda _j s + \mu _j), \end{aligned}$$

  with $Q_F> 0, r \ge 0, \lambda _j> 0, |w|=1, \mathfrak {R}(\mu _j)> - \frac{1}{4}, \mathfrak {R}(\lambda _j + 2 \mu _j) > 0, j = 1, \ldots , r$.

• (5’) (Euler sum) The logarithmic derivative of the function F possesses a Dirichlet series representation

  $$\begin{aligned} \frac{F'}{F}(s)= - \sum _{n=1}^{\infty }\frac{c_F(n)}{n^s}, \end{aligned}$$

  converging absolutely for $\mathfrak {R}s > 1$.

Let us note that (3’) implies that $\mathfrak {R}(\lambda _j + \mu _j) > 0$. If $F \in \mathcal {S}$ then

$$\begin{aligned} c_F(n) = b_F(n) \log n. \end{aligned}$$

(2)

Assuming GRH, we give an upper bound for the multiplicity of eventual zero at central point 1 / 2. Moreover, we give a bound for the location of the first zero with positive imaginary part of function F in $\mathcal {S}^{\sharp \flat }$ such that $\mathfrak {R}(c_F(n)) \ge 0$ for all $n \in \mathbb {N}$.

Similar results for Dedekind zeta function were obtained in  [11].

The paper is organized as follows. In Sect. 2 we will give the main results of the paper. In Sect. 3 we recall an explicit formula we use in the proof of our main results. In Sect. 4 we prove preliminary lemmas. In Sect. 5 we prove main results of the paper. In Sect. 6 we apply results of Sect. 2 to automorphic L-functions attached to irreducible unitary automorphic representations of $GL_N(\mathbb {Q})$.

2 Main Results

In this section we give two main results of the paper. Namely, we give an upper bound for the multiplicity of eventual zero at central point 1 / 2 and provide an upper bound for the height of the first zero with positive imaginary part of function F in $\mathcal {S}^{\sharp \flat }$ such that $\mathfrak {R}(c_F(n)) \ge 0$ for all $n \in \mathbb {N}$.

Throughout this section we assume the GRH i.e. we assume that all non-trivial zeros of $F \in \mathcal {S}^{\sharp \flat }$ are on the line $\mathfrak {R}s = 1/2$.

2.1 Multiplicity of Eventual Zero at Central Point

Theorem 1

Let R be the multiplicity of eventual zero at central point 1 / 2 of function $F \in \mathcal {S}^{\sharp \flat }$ such that $\mathfrak {R}({c_F} (n)) \ge 0$ and let $B(F) = 2\sum _{j=1}^{r} \lambda _j \Big ( \mathfrak {R}\Big ( {\varPsi }\Big ( \frac{\lambda _j}{2} + \mu _j \Big ) \Big ) - \log (2\pi \lambda _j) \Big )$.

• (a) If $q_F > e$ then

  $$\begin{aligned} R \le \frac{(4m_F + 1)\log q_F + B(F)}{2 \log \log q_F} . \end{aligned}$$

• (b) If $0 < q_F \le e$ then

  • (i) $R=0$, for $m_F = 0$,

  • (ii) $R \le \frac{4m_F e^{ W \big ( \frac{B(F) + 1}{4 e m_F}\big ) + 1} + B(F) + 1}{2 \Big ( W \Big ( \frac{B(F) + 1}{4 e m_F}\Big ) + 1\Big )}$, for $4m_F + B(F) + 1 > 0$,

where $m_F$ is the polar order of $F, q_F$ is the conductor of $F, \lambda _j, \mu _j$ are given as in axiom (3’) and W denotes the Lambert function.

2.2 Location of the First Zero with Positive Imaginary Part

Theorem 2

  Let h be the height of the first zero with imaginary part different from zero of the function $F \in \mathcal {S}^{\sharp \flat }$. Assume that F satisfies axiom (5) of the Selberg class and $\mathfrak {R}({c_F} (n)) \ge 0$. Then, for $q_F > e$ we have the bound

$$\begin{aligned} h \le \max \left\{ \frac{16 \sqrt{2}\Big [(4m_F + 1)\log q_F + B(F) \Big ]}{\pi \log q_F \log \log q_F}, \frac{(2 \theta + 1) \pi }{\sqrt{2} \log [\log q_F / 16(K_F + \delta ) ]} \right\} . \end{aligned}$$

Here $q_F$ is the conductor of $F, m_F$ is the polar order of F, B(F) is given in Theorem 1, $K_F$ is defined in Lemma 3, $\theta < 1/2$ stemmed from axiom (5) of the Selberg class and $\delta > 0$.

In the case when $F\in \mathcal {S}$ with non-negative coefficients, we can get sharper upper bound for the height of the first zero of F with positive imaginary part, as stated in the following

Theorem 3

Let h be the height of the first zero with imaginary part different from zero of the function $F \in \mathcal {S}$ and $F(1+it) \ne 0$ for all $t \in \mathbb {R}$ such that $a_F(n)\ge 0$ for all $n\in \mathbb {N}$. Then, for $q_F > e$ we have the bound

$$\begin{aligned} h \le \max \left\{ \frac{16 \sqrt{2}\Big [(4m_F + 1)\log q_F + B(F) \Big ]}{\pi \log q_F \log \log q_F}, \frac{\pi }{\sqrt{2} \log [\log q_F / 16(m_F + \tau ) ]} \right\} , \end{aligned}$$

where $q_F$ is conductor of F, $m_F$ is the polar order of F, B(F) is given in Theorem 1 and $\tau > 0$.

3 Preliminaries

3.1 Explicit Formula for Functions in $\mathcal {S}^{\sharp \flat }$

The universal class of test functions in this paper is the class W of regulated functions  [3] i.e. functions possessing the one-sided limits at each point. For $f \in W$, we always suppose $2f(x)=f(x+0)+f(x-0)$. If I is an interval with endpoints a and $b (a<b)$, we write $f(I)=f(b)-f(a).$

Let $\phi$ be continuous function defined on $[0,\infty )$ and strictly increasing from 0 to $\infty$. A function f is said to be of $\phi$-bounded variation on I if

$$\begin{aligned} V_{\phi }(f,I)=\sup \sum _{n}\phi (|f(I_{n})|), \end{aligned}$$

where the supremum is taken over all systems $\{I_n\}$ of non-overlapping subintervals of I (cf.  [22]).

The crucial tool for deriving our main results is the explicit formula for functions in the Selberg class and its generalizations, applied to suitably constructed test functions.

Theorem 4

 [20, Theorem 3.1], [10, Proposition 2.2] Let a regularized function G satisfy the following conditions:

• 1. $G \in \phi BV(\mathbb {R})\cap L^{1}(\mathbb {R}).$

• 2. $G(x)e^{(1/2+\epsilon )|x|} \in \phi BV(\mathbb {R})\cap L^{1}(\mathbb {R})$, for some $\epsilon > 0.$

• 3. $G(x) + G(-x) - 2 G(0) = O(|\log |x||^{-\alpha })$, as $x \rightarrow 0$, for some $\alpha > 2.$

Let $g(x)=G(-\log x)$, for $x>0, G_j(x)=G(x)\exp \Big (\frac{ix\mathfrak {I}\mu _j}{\lambda _j} \Big )$ and Z(F) the set of all non-trivial zeros of $F\in \mathcal {S}^{\sharp \flat }$. Then, the formula

$$\begin{aligned}&\lim _{a \rightarrow \infty } \sum _{\begin{array}{c} \rho \in Z(F) |\mathfrak {I}\rho | \le a \end{array}} ord (\rho ) M_{\frac{1}{2}} g(\rho ) \\&\quad = m_F M_{\frac{1}{2}} g(0) + m_F M_{\frac{1}{2}} g(1) \\&\quad \quad - \sum _{n} \frac{c_F (n)}{n^{\frac{1}{2}}} g(n) - \sum _{n} \frac{\overline{c_F} (n)}{n^{\frac{1}{2}}} g(1/n) + 2 G(0) \log Q_F \\&\quad \quad + \sum _{j=1}^{r} \int \limits _{0}^{\infty } \Bigg [ \frac{2\lambda _j G_j(0)}{x} - \frac{\exp ((1-\frac{\lambda _j}{2} - \mathfrak {R}\mu _j) \frac{x}{\lambda _j})}{1 - e^{\frac{-x}{\lambda _j}}} (G_j(x) + G_j(-x))\Bigg ] e^{\frac{-x}{\lambda _j}} \mathrm{d}x \end{aligned}$$

holds true for an arbitrary function $F \in \mathcal {S}^{\sharp \flat }$, where

$$\begin{aligned} M_{\frac{1}{2}} g(s) = \int \limits _{-\infty }^{\infty } G(x) e^{(s-1/2)x}\mathrm{d}x \end{aligned}$$

denotes the translate by 1 / 2 of the Mellin transform of the function g.

Corollary 1

Let G be an even regularized function satisfying conditions of Theorem 4 then, the formula

$$\begin{aligned}&\lim _{a \rightarrow \infty } \sum _{\begin{array}{c} \rho \in Z(F) |\mathfrak {I}\rho | \le a \end{array}} ord (\rho ) M_{\frac{1}{2}} g(\rho )\nonumber \\= & {} m_F M_{\frac{1}{2}} g(0) + m_F M_{\frac{1}{2}} g(1) - 2\sum _{n} \frac{\mathfrak {R}( {c_F} (n))}{n^{\frac{1}{2}}} g(1/n) \nonumber \\+ & {} G(0)\Big ( \log q_F - d_F \log (2 \pi ) - 2\sum _{j=1}^{r} (\lambda _j \log \lambda _j) \Big ) \nonumber \\+ & {} 2 \sum _{j=1}^{r} \int \limits _{0}^{\infty } \Bigg [ \frac{\lambda _j G(0)}{x} - \frac{\exp ((1-\frac{\lambda _j}{2} - \mathfrak {R}\mu _j) \frac{x}{\lambda _j})}{1 - e^{\frac{-x}{\lambda _j}}} G(x) \cosh \Big ( \frac{ix\mathfrak {I}\mu _j}{\lambda _j} \Big )\Bigg ] e^{\frac{-x}{\lambda _j}} \mathrm{d}x\nonumber \\ \end{aligned}$$

(3)

holds true for an arbitrary function $F \in \mathcal {S}^{\sharp \flat }$.

Proof

Since G is even function then

$$\begin{aligned} G(-\log x) = G(\log x), \quad x>0, \end{aligned}$$

hence $g(x) = g(1/x)$, which yields

$$\begin{aligned} \sum _{n} \frac{c_F (n)}{n^{\frac{1}{2}}} g(n) + \sum _{n} \frac{\overline{c_F} (n)}{n^{\frac{1}{2}}} g(1/n) = 2\sum _{n} \frac{\mathfrak {R}( {c_F} (n))}{n^{\frac{1}{2}}} g(1/n), \end{aligned}$$

and

$$\begin{aligned} G_j(x) + G_j(-x) = 2 G(x) \cosh \Big ( \frac{ix\mathfrak {I}\mu _j}{\lambda _j} \Big ). \end{aligned}$$

Furthermore, from (1) we get

$$\begin{aligned} 2 \log Q_F = \log q_F - d_F \log (2 \pi ) - 2\sum _{j=1}^{r} \lambda _j \log \lambda _j. \end{aligned}$$

This completes the proof. $\square$

3.2 The Prime Number Theorem in the Selberg Class

For $F \in \mathcal {S}$ let us denote by

$$\begin{aligned} \psi _F(x) = \sum _{n \le x} c_F(n) \end{aligned}$$

the analogue of the Chebyshev $\psi$-function, where $c_F(n)$ is defined by (2).

The Selberg class analogue of the prime number theorem is a theorem that explains the asymptotic behaviour of the function $\psi _F(x)$, as $x \rightarrow \infty$.

Kaczorowski and Perelli  [9] have proved the equivalence between the prime number theorem for the Selberg class and non-vanishing on the line $\mathfrak {R}s = 1$ for every function in $\mathcal {S}$, without using Tauberian arguments. They proved the following theorem.

Theorem 5

 [9, Theorem 1] Let $F \in \mathcal {S}$. Then $\psi _F(x) = m_F x + p_F(x)$, where $p_F(x)=o(x)$ as $x \rightarrow \infty$ if and only if $F(1 + it) \ne 0$ for every $t\in \mathbb {R}$.

4 Preliminary Lemmas

In the proof of our main results, we will need the following lemmas.

Lemma 1

 [11, p. 63] Let G be defined by

$$\begin{aligned} G(x) = \left\{ \begin{array}{ll} 1 - |x|, &{} \quad \mathrm{if} \; |x| \le 1 \\ 0, &{} \quad \mathrm{otherwise}. \end{array} \right. \end{aligned}$$

Then G satisfies the conditions of Corollary  1 and

$$\begin{aligned} \hat{G}(u) = \Bigg ( \frac{2 \sin \frac{u}{2}}{u} \Bigg )^{2}, \end{aligned}$$

where $\hat{G}$ is the Fourier transform of G.

Lemma 2

 [11, Lemma 1] Let H be the function with compact support on $[0,\infty ]$ defined by

$$\begin{aligned} H(x) = \left\{ \begin{array}{ll} (1 - x) \cos (\pi x) + \frac{3}{\pi } \sin (\pi x), &{} \quad \mathrm{if} \; 0 \le x \le 1, \\ (1 + x) \cos (\pi x) - \frac{3}{\pi } \sin (\pi x), &{} \quad \mathrm{if} \; -1 \le x < 0, \\ 0, &{} \quad \mathrm{otherwise}. \end{array} \right. \end{aligned}$$

Then H satisfies the condition of Corollary 1 and

$$\begin{aligned} \hat{H}(u) = 2 \Bigg (2 - \frac{u^2}{\pi ^2} \Bigg ) \Bigg ( \frac{2 \pi }{\pi ^2 - u^2} \cos \frac{u}{2} \Bigg )^{2}. \end{aligned}$$

The proof of Lemmas 1 and 2 is based on partial integration of the Mellin transform.

Let $b_F(n)$ be as in axiom (5) of the Selberg class. Then there exists $C_F\ge 1$ such that

$$\begin{aligned} |b_F(n)| \le C_F n^{\theta }, \,\,\, \theta < 1/2. \end{aligned}$$

(4)

The Chebyshev function is defined by $\psi (x) = \sum _{n \le x} {\varLambda }(n)$, where ${\varLambda }(n)$ is von Mangoldt function. It satisfies the asymptotic formula

$$\begin{aligned} \psi (x) = x + r(x), \end{aligned}$$

(5)

where $r(x) = O(x \exp (-a \sqrt{\log x}))$ for some $a>0$ and x large enough (see e.g.  [2, p.111]).

Lemma 3

  Let $H_T(x ) = H (x / T)$, where H is defined in Lemma 2 and $g_T(1/n) = H \Big ( \frac{\log n}{T} \Big )$.

• (a) For $F \in \mathcal {S^{\sharp \flat }}$ satisfying axiom (5) of the Selberg class we have

  $$\begin{aligned} \sum _{n} \frac{|c_F (n)|}{n^{\frac{1}{2}}} g_T(1/n) \le 4K_F e^{\frac{T}{2}(2 \theta + 1)} + r_2(T), \end{aligned}$$

  where

  $$\begin{aligned} K_F = \frac{C_F}{2\theta + 1}, \end{aligned}$$
  $$\begin{aligned} r_2(T)= C_F \Bigg ( 2 e^{\frac{T}{2}(2 \theta - 1)}r(e^T)+ 2 r_1(e^T)e^{-\frac{T}{2}} + \int \limits _{1}^{e^{T}} x^{\theta - \frac{3}{2}}r(x) dx + \int \limits _{1}^{e^{T}} x^{ - \frac{3}{2}}r_1(x) \mathrm{d}x \Bigg ), \end{aligned}$$
  $$\begin{aligned} r_1(x)= \frac{\theta }{\theta + 1} - \theta \int \limits _{1}^{x} t^{\theta - 1}r(t) \mathrm{d}t, \end{aligned}$$

  and $C_F, r(x)$ are as in (4), (5), respectively.

• (b) For $F \in \mathcal {S}$ and $F(1+it) \ne 0$ such that $a_F(n)\ge 0$ for all $n\in \mathbb {N}$ we have

  $$\begin{aligned} \sum _{n} \frac{c_F (n)}{n^{\frac{1}{2}}} g_T(1/n) \le 4 m_F e^{\frac{T}{2}} + P_F(T), \end{aligned}$$

  where

  $$\begin{aligned} P_F(T) = 2 p_F(e^{T})e^{-\frac{T}{2}} + \int \limits _{1}^{e^{T}} p_F(x)x^{-\frac{3}{2}} \mathrm{d}x , \end{aligned}$$

  $m_F$ is as in axiom (2) of the Selberg class and $p_F$ is as in Theorem 5.

Proof

Definition of H yields

$$\begin{aligned} H \left( \frac{\log n}{T} \right) = \left\{ \begin{array}{ll} \left( 1 - \frac{\log n}{T}\right) \cos \left( \frac{ \pi \log n}{T} \right) + \frac{3}{\pi } \sin \left( \frac{ \pi \log n}{T} \right) , &{} \quad \mathrm{if} \; 0 \le \log n \le T, \\ \left( 1 + \frac{\log n}{T}\right) \cos \left( \frac{ \pi \log n}{T} \right) - \frac{3}{\pi } \sin \left( \frac{ \pi \log n}{T} \right) , &{} \quad \mathrm{if} \; -T \le \log n < 0, \\ 0, &{} \quad \mathrm{otherwise}, \end{array} \right. \end{aligned}$$

hence

$$\begin{aligned} - 2 \le H \left( \frac{\log n}{T} \right) \le 2, \,\,\, \text {for} \, \, \, e^{-T} \le n\le e^{T}. \end{aligned}$$

• (a) Let $\varphi _F(x) = \sum _{n\le x} |c_F (n)|$. From (2) and (4) we get

  $$\begin{aligned} \varphi _F(x) = \sum _{n \le x} |b_F (n)| \log n \le C_F \sum _{p^k \le x} p^{k \theta } \log p^{k} = C_F \sum _{n \le x} n^{\theta } {\varLambda }(n). \end{aligned}$$

  Therefore

  $$\begin{aligned} \varphi _F(x) \le C_F \sum _{n \le x} n^{\theta } {\varLambda }(n) = C_F \int \limits _{1}^{x} t^{\theta } d \psi (t). \end{aligned}$$

  With partial integration we have

  $$\begin{aligned} \int \limits _{1}^{x} t^{\theta } d \psi (t) = \frac{1}{\theta + 1}x^{\theta + 1 } + x^{\theta } r(x) + r_1(x) , \end{aligned}$$

  hence

  $$\begin{aligned} \varphi _F(x) \le \frac{C_F}{\theta + 1} x^{\theta +1} + C_F x^{\theta } r(x) + C_F r_1(x). \end{aligned}$$

  Now, we have the following estimate of the sum

  $$\begin{aligned} \sum _{n} \frac{|b_F (n)| \log n}{n^{\frac{1}{2}}} g_T(1/n) \le 2 \sum _{n \le e^T} \frac{|b_F (n)| \log n}{n^{\frac{1}{2}}} = 2\int \limits _{1}^{e^T} \frac{1}{x^{1/2}} d \varphi _F(x). \end{aligned}$$

  An integration by parts of the last integral gives

  $$\begin{aligned} \int \limits _{1}^{e^T} \frac{1}{x^{1/2}} d \varphi _F(x) \le \frac{\varphi _F(e^{T})}{e^{T/2}} + \frac{1}{2} \int \limits _{1}^{e^{T}} \frac{\varphi _F(x)}{x^{3/2}} \mathrm{d}x \le 2 K_F e^{\frac{T}{2}(2 \theta + 1)} + \frac{1}{2} r_2(t) , \end{aligned}$$

  it follows

  $$\begin{aligned} \sum _{n} \frac{|b_F (n)| \log n}{n^{\frac{1}{2}}} g_T(1/n) \le 4 K_F e^{\frac{T}{2}(2 \theta + 1)} + r_2(t) . \end{aligned}$$

• (b) Since $F \in \mathcal {S}$ and $F(1+it) \ne 0$ from (2), Theorem 5 and definition of $g_T$ we have

  $$\begin{aligned} \sum _{n} \frac{c_F (n)}{n^{\frac{1}{2}}} g_T(1/n) \le 2 \sum _{1 \le n \le e^T} \frac{c_F(n)}{n^{\frac{1}{2}}} = 2 \int \limits _{1}^{e^T} \frac{1}{x^{\frac{1}{2}}} d \psi _F(x). \end{aligned}$$

  With partial integration we have

  $$\begin{aligned} \int \limits _{1}^{e^T} \frac{1}{x^{\frac{1}{2}}} d \psi _F(x) \le 2 m_F e^{\frac{T}{2}} + \frac{1}{2}P_F(T), \end{aligned}$$

  hence

  $$\begin{aligned} \sum _{n} \frac{c_F (n)}{n^{\frac{1}{2}}} g_T(1/n) \le 4 m_F e^{\frac{T}{2}} + P_F(T). \end{aligned}$$

$\square$

Lemma 4

   [11, Lemma 3] Let A, B, C be three positive real constants and $\alpha > 0$. If $T > 0$ satisfies $AT + B e^{\alpha T} \ge C$, then

$$\begin{aligned} T \ge \min \Bigg \{ \frac{C}{2A}, \frac{ \log (C/2B)}{\alpha } \Bigg \} . \end{aligned}$$

Proof

By contradiction. $\square$

5 Proof of Main Results

In this section we prove main results of the paper given in Sect. 2.

5.1 Proof of Theorem 1

Let $s = \sigma + it$. The Mellin transform of G is given by

$$\begin{aligned} M_{\frac{1}{2}} g(s) = \int \limits _{-\infty }^{\infty } G(x) e^{(s-1/2)x}\mathrm{d}x = \int \limits _{-\infty }^{\infty } G(x) e^{(\sigma -1/2)x} e^{itx} \mathrm{d}x = \hat{G}_{\sigma }(t), \end{aligned}$$

where

$$\begin{aligned} G_{\sigma }(t) = G(x) e^{(\sigma -1/2)x}. \end{aligned}$$

If $\sigma = 1/2$ then

$$\begin{aligned} M_{\frac{1}{2}} g \Big (\frac{1}{2}+it \Big ) = \int \limits _{-\infty }^{\infty } G(x) e^{itx} \mathrm{d}x = \hat{G}(t) \end{aligned}$$

For $t=0$ we have

$$\begin{aligned} M_{\frac{1}{2}} g \Big (\frac{1}{2} \Big ) = \hat{G}(0) = 1. \end{aligned}$$

Now,

$$\begin{aligned} M_{\frac{1}{2}} g(0) + M_{\frac{1}{2}} g(1) = \int \limits _{-\infty }^{\infty } G(x) e^{-\frac{x}{2}} \mathrm{d}x + \int \limits _{-\infty }^{\infty } G(x) e^{\frac{x}{2}} \mathrm{d}x = 4 \int \limits _{0}^{\infty } G(x) \cosh \Big ( \frac{x}{2} \Big ) \mathrm{d}x. \end{aligned}$$

Setting $G_T(x) = G(x/T)$ for $T>0$ we get

$$\begin{aligned} \hat{G}_{T}(u) = \int \limits _{-\infty }^{\infty } G_T(x) e^{iux} \mathrm{d}x = \int \limits _{-\infty }^{\infty } G(x/T) e^{iux} \mathrm{d}x. \end{aligned}$$

Substituting $x/T = t$ we get

$$\begin{aligned} \hat{G}_{T}(u) = T \int \limits _{-\infty }^{\infty } G(t) e^{iuTt} \mathrm{d}t = T \hat{G}(Tu). \end{aligned}$$

If R is order of eventual zero of $F(s) \in \mathcal {S}^{\sharp \flat }$ at point $\rho = 1/2$ then applying explicit formula (3) for the function $G_T(x)$ we obtain the inequalities

$$\begin{aligned}&R M_{\frac{1}{2}} g_T(1/2)\nonumber \\&\quad \le \lim _{a \rightarrow \infty } \sum _{\begin{array}{c} \rho \in Z(F) |\mathfrak {I}\rho | \le a \end{array}} ord (\rho ) M_{\frac{1}{2}} g_T(\rho )\nonumber \\&\quad = m_F ( M_{\frac{1}{2}} g_T(0) + M_{\frac{1}{2}} g_T(1)) - 2\sum _{n} \frac{\mathfrak {R}( {c_F} (n))}{n^{\frac{1}{2}}} g_T(1/n)\nonumber \\&\qquad + G_T(0)\Big ( \log q_F - d_F \log (2 \pi ) - 2\sum _{j=1}^{r} (\lambda _j \log \lambda _j) \Big ) \nonumber \\&\qquad + 2 \sum _{j=1}^{r} \int \limits _{0}^{\infty } \Bigg [ \frac{\lambda _j G_{T, j}(0)}{x} - \frac{\exp ((1-\frac{\lambda _j}{2} - \mathfrak {R}(\mu _j)) \frac{x}{\lambda _j})}{1 - e^{\frac{-x}{\lambda _j}}} G_T(x) \cosh \Big ( \frac{ix\mathfrak {I}\mu _j}{\lambda _j} \Big )\Bigg ] e^{\frac{-x}{\lambda _j}} \mathrm{d}x \nonumber \\&\quad \le 4 m_F \int \limits _{0}^{\infty } G_T(x) \cosh \Big ( \frac{x}{2} \Big ) \mathrm{d}x + \log q_F -2 \sum _{j=1}^{r}\lambda _j \log (2 \pi \lambda _j))\nonumber \\&\qquad + \sum _{j=1}^{r} \int \limits _{0}^{\infty } \Bigg [ \frac{2\lambda _j G_{T, j}(0)}{x} - \frac{\exp ((1-\frac{\lambda _j}{2} - \mathfrak {R}(\mu _j)) \frac{x}{\lambda _j})}{1 - e^{\frac{-x}{\lambda _j}}} (G_{T, j}(x) + G_{T, j}(-x))\Bigg ] e^{\frac{-x}{\lambda _j}} \mathrm{d}x.\nonumber \\ \end{aligned}$$

(6)

We denote by

$$\begin{aligned} I = \int \limits _{0}^{\infty } \Bigg [ \frac{2\lambda _j G_{T, j}(0)}{x} - \frac{\exp ((1-\frac{\lambda _j}{2} - \mathfrak {R}\mu _j) \frac{x}{\lambda _j})}{1 - e^{\frac{-x}{\lambda _j}}} (G_{T, j}(x) + G_{T, j}(-x))\Bigg ] e^{\frac{-x}{\lambda _j}} \mathrm{d}x. \end{aligned}$$

Substituting in the above integral $\frac{x}{\lambda _j} = t$, employing the equality $G_{T,j}(0) = 1$, we get

$$\begin{aligned} I = \lambda _j \int \limits _{0}^{\infty } \Bigg [ \frac{2}{t} - \frac{\exp ((1-\frac{\lambda _j}{2} - \mathfrak {R}\mu _j) t)}{1 - e^{-t}} (G_{T, j}(\lambda _j t) + G_{T, j}(-\lambda _j t))\Bigg ] e^{-t} \mathrm{d}t. \end{aligned}$$

By the definition of function $G_j$ it follows that

$$\begin{aligned} G_{T, j}(\lambda _j t) + G_{T, j}(-\lambda _j t) = \left\{ \begin{array}{ll} \Big ( 1 - \frac{\lambda _j t}{T} \Big ) \Big ( e^{it \mathfrak {I}\mu _j } + e^{- it \mathfrak {I}\mu _j } \Big ), &{} \quad \mathrm{if} \; 0 \le t \le \frac{T}{\lambda _j} \\ 0, &{} \quad \mathrm{otherwise}. \end{array} \right. \end{aligned}$$

For $0 \le t \le \frac{T}{\lambda _j}$

$$\begin{aligned} \Big ( 1 - \frac{\lambda _j t}{T} \Big ) \le 1, \end{aligned}$$

hence

$$\begin{aligned} I&\le \lambda _j \Bigg [ \int \limits _{0}^{\infty } \Bigg [ \frac{1}{t} - \frac{\exp ((1-\frac{\lambda _j}{2} - \mathfrak {R}\mu _j) t)}{1 - e^{-t}} e^{it \mathfrak {I}\mu _j } \Bigg ] e^{-t} \mathrm{d}t \\&+ \int \limits _{0}^{\infty } \Bigg [ \frac{1}{t} - \frac{\exp ((1-\frac{\lambda _j}{2} - \mathfrak {R}\mu _j) t)}{1 - e^{-t}} e^{- it \mathfrak {I}\mu _j } \Bigg ] e^{-t} \mathrm{d}t \Bigg ]\\&= \lambda _j \Bigg [ \int \limits _{0}^{\infty } \Bigg [ \frac{1}{t} - \frac{\exp ((1-\frac{\lambda _j}{2} - \bar{\mu }_j) t)}{1 - e^{-t}} \Bigg ] e^{-t} \mathrm{d}t \\&+ \int \limits _{0}^{\infty } \Bigg [ \frac{1}{t} - \frac{\exp ((1-\frac{\lambda _j}{2} - \mu _j) t)}{1 - e^{-t}} e^{- it \mathfrak {I}\mu _j } \Bigg ] e^{-t} \mathrm{d}t \\&= \lambda _j \bigg ( {\varPsi }\Big ( \frac{\lambda _j}{2} + \bar{\mu }_j \Big ) + {\varPsi }\Big ( \frac{\lambda _j}{2} + \mu _j \Big ) \bigg ). \end{aligned}$$

Since

$$\begin{aligned} 4 \int \limits _{0}^{\infty } G_T(x) \cosh \Big ( \frac{x}{2} \Big ) \mathrm{d}x&= 4 \int \limits _{0}^{T} (1 - \frac{x}{T}) \cosh \Big ( \frac{x}{2} \Big ) \mathrm{d}x \\&\le 4 \int \limits _{0}^{T} \cosh \Big ( \frac{x}{2} \Big ) \mathrm{d}x \le 4 e^{T/2}, \end{aligned}$$

$$\begin{aligned} M_{\frac{1}{2}} g_T(1/2) = T \hat{G}(T \cdot 0) = T \cdot 1 = T, \end{aligned}$$

we get

$$\begin{aligned} R T\le & {} 4 m_F e^{T/2} + \log q_F -2 \sum _{j=1}^{r}\lambda _j \log (2 \pi \lambda _j) \\&+\, \sum _{j=1}^{r} \lambda _j \bigg ( {\varPsi }\Big ( \frac{\lambda _j}{2} + \bar{\mu }_j \Big ) \bigg ) +\sum _{j=1}^{r} \lambda _j \bigg ( {\varPsi }\Big ( \frac{\lambda _j}{2} + \mu _j \Big ) \bigg ). \end{aligned}$$

It follows that

$$\begin{aligned} R T\le & {} 4 m_F e^{T/2} + \log q_F -2 \sum _{j=1}^{r}\lambda _j \log (2 \pi \lambda _j) + 2 \sum _{j=1}^{r} \lambda _j \mathfrak {R}\Big ( {\varPsi }\Big ( \frac{\lambda _j}{2} + \mu _j \Big ) \Big ) \\= & {} 4 m_F e^{T/2} + \log q_F + B(F). \end{aligned}$$

Setting $T= 2\log \log q_F$ for $q_F > e$ we get

$$\begin{aligned} 2 R \log \log q_F \le 4 m_F \log q_F + \log q_F + B(F), \end{aligned}$$

hence

$$\begin{aligned} R \le \frac{ 4 m_F \log q_F + \log q_F + B(F)}{2\log \log q_F }. \end{aligned}$$

If $0 < q_F \le e$, then

$$\begin{aligned} R T \le 4 m_F e^{T/2} + B(F) +1 , \end{aligned}$$

hence

$$\begin{aligned} R \le \inf _{T>0} \Big \{ \frac{ 4 m_F e^{T/2} + B(F) +1 }{T} \Big \}. \end{aligned}$$

If $m_F=0$ then $\inf \limits _ {T>0} \Big \{ \frac{ 4 m_F e^{T/2} + B(F) +1 }{T} \Big \} = 0$. Otherwise, let

$$\begin{aligned} f(T) = \frac{ 4 m_F e^{T/2} + B(F) +1 }{T}. \end{aligned}$$

Then function f has minimum at point $T>0$ satisfying equation

$$\begin{aligned} 2 m_F e^{T/2}(T-2) = B(F) + 1. \end{aligned}$$

We can solve the last equation using the Lambert W-function and get

$$\begin{aligned} T = 2 \Big ( W \Big ( \frac{B(F) + 1}{4 e m_F}\Big ) + 1\Big ). \end{aligned}$$

This proves our theorem.

As an immediate consequence of the above theorem, in the case when the conductor of function F is small, we get the following

Corollary 2

Let $F \in \mathcal {S}^{\sharp \flat }$ be such that $\mathfrak {R}({c_F} (n)) \ge 0$. Assume also that the conductor, $q_F$ of F is less then or equal to e and that F is holomorphic. Then, $F(1/2) \ne 0$, i.e. F is non-vanishing at the central point.

Remark 1

From the proof of the Theorem 1 it is easy to see that the statement of theorem holds true under slightly less restrictive assumptions on $\mathfrak {R}({c_F} (n))$. Namely, it is sufficient to assume that

$$\begin{aligned} \sum _{n} \frac{\mathfrak {R}( {c_F} (n))}{n^{\frac{1}{2}}} g_T(1/n) \ge 0, \end{aligned}$$

see formula (6).

5.2 Proof of Theorem 2

For $T = \sqrt{2} \pi / h$ and $u\ge h$ it is easy to see that

$$\begin{aligned} M_{\frac{1}{2}} g_T\Big (\frac{1}{2} +iu \Big ) = \hat{H}_T(u) \le 0, \end{aligned}$$

hence from the GRH and Lemma 2 we have

$$\begin{aligned} \sum _{\begin{array}{c} \rho \in Z(F) |\mathfrak {I}\rho | \le a \end{array}} ord (\rho ) M_{\frac{1}{2}} g_T(\rho )= & {} R M_{\frac{1}{2}} g_T(0) + \sum _{\begin{array}{c} \rho \in Z(F) \rho \ne 1/2 |\mathfrak {I}\rho | \le a \end{array}} ord (\rho ) M_{\frac{1}{2}} g_T(\rho ) \\\le & {} R \hat{H}_T(0)=\frac{16}{\pi ^2}RT, \end{aligned}$$

for all $a>1$. Therefore letting $a \rightarrow \infty$ and applying explicit formula (3) we obtain the inequality

$$\begin{aligned}&\frac{16}{\pi ^2} R T \ge m_F ( M_{\frac{1}{2}} g_T(0) + M_{\frac{1}{2}} g_T(1)) - 2\sum _{n} \frac{\mathfrak {R}( {c_F} (n))}{n^{\frac{1}{2}}} g_T(1/n) \nonumber \\&\quad + H_T(0)\Big ( \log q_F - d_F \log (2 \pi ) - 2\sum _{j=1}^{r} (\lambda _j \log \lambda _j) \Big ) \nonumber \\&\quad + \sum _{j=1}^{r} \int \limits _{0}^{\infty } 2 \Bigg [ \frac{\lambda _j H_{T, j}(0)}{x} - \frac{\exp ((1-\frac{\lambda _j}{2} - \mathfrak {R}\mu _j) \frac{x}{\lambda _j})}{1 - e^{\frac{-x}{\lambda _j}}} H_T(x) \cosh \Big ( \frac{ix\mathfrak {I}\mu _j}{\lambda _j} \Big )\Bigg ] e^{\frac{-x}{\lambda _j}} \mathrm{d}x.\nonumber \\ \end{aligned}$$

(7)

Since

$$\begin{aligned} M_{\frac{1}{2}} g_T(0) + M_{\frac{1}{2}} g_T(1) = 4 \int \limits _{0}^{\infty } H_T(x) \cosh \Big ( \frac{x}{2} \Big ) \mathrm{d}x, \end{aligned}$$

by the definition of function $H_T(x)$ we have

$$\begin{aligned} M_{\frac{1}{2}} g_T(0) + M_{\frac{1}{2}} g_T(1) = 4 \int \limits _{0}^{T} \Bigg [ \Big ( 1 - \frac{x}{T}\Big ) \cos \Big ( \frac{\pi x}{T} \Big ) + \frac{3}{\pi } \sin \Big ( \frac{\pi x}{T} \Big ) \Bigg ] \cosh \Big ( \frac{x}{2} \Big ) \mathrm{d}x. \end{aligned}$$

Using partial integration we get

$$\begin{aligned} M_{\frac{1}{2}} g_T(0) + M_{\frac{1}{2}} g_T(1) \ge \frac{16 T^3}{(4 \pi ^2 + T^2)^2} e^{\frac{T}{2}}. \end{aligned}$$

(8)

Since $H_{T, j}(0) = 1$ and from the definition of the function $H_T(x)$ we get

$$\begin{aligned}&\int \limits _{0}^{\infty } 2 \left[ \frac{\lambda _j H_{T, j}(0)}{x} - \frac{\exp ((1-\frac{\lambda _j}{2} - \mathfrak {R}\mu _j) \frac{x}{\lambda _j})}{1 - e^{\frac{-x}{\lambda _j}}} H_T(x) \cosh \left( \frac{ix\mathfrak {I}\mu _j}{\lambda _j} \right) \right] e^{\frac{-x}{\lambda _j}} \mathrm{d}x \\&\quad = \int \limits _{0}^{T} \left[ \frac{\lambda _j e^{\frac{-x}{\lambda _j}}}{x} - \frac{e^{-(\frac{\lambda _j}{2} + \mu _j)\frac{x}{\lambda _j}}}{1 - e^{\frac{-x}{\lambda _j}}}\left( \left( 1 - \frac{x}{T} \right) \cos \left( \frac{\pi x}{T} \right) + \frac{3}{\pi } \sin \left( \frac{\pi x}{T} \right) \right) \right] \mathrm{d}x \\&\qquad + \int \limits _{0}^{T} \left[ \frac{\lambda _j e^{\frac{-x}{\lambda _j}}}{x} - \frac{e^{-(\frac{\lambda _j}{2} + \bar{\mu }_j)\frac{x}{\lambda _j}}}{1 - e^{\frac{-x}{\lambda _j}}} \left( \left( 1 - \frac{x}{T} \right) \cos \left( \frac{\pi x}{T} \right) + \frac{3}{\pi } \sin \left( \frac{\pi x}{T} \right) \right) \right] \mathrm{d}x\\&\quad = I_1 + I_2. \end{aligned}$$

For simplicity, we evaluate $I_1$. Substituting $x=tT$ we have

$$\begin{aligned} I_1 = T \int \limits _{0}^{1} \left[ \frac{ e^{\frac{-Tt}{\lambda _j}}}{\frac{Tt}{\lambda _j}} - \frac{e^{-(\frac{\lambda _j}{2} + \mu _j)\frac{Tt}{\lambda _j}}}{1 - e^{\frac{-Tt}{\lambda _j}}} \left( ( 1 - t ) \cos ( \pi t ) + \frac{3}{\pi } \sin ( \pi t ) \right) \right] \mathrm{d}x. \end{aligned}$$

Expanding the function under the integral sign in the Taylor series at $t=0$, we see that it is bounded as $t \rightarrow 0$ and the bound is independent of T, hence

$$\begin{aligned}&\sum _{j=1}^{r} \int \limits _{0}^{\infty } 2 \left[ \frac{\lambda _j H_{T, j}(0)}{x} - \frac{\exp \left( \left( 1-\frac{\lambda _j}{2} - \mathfrak {R}\mu _j \right) \frac{x}{\lambda _j} \right) }{1 - e^{\frac{-x}{\lambda _j}}} H_T(x) \cosh \left( \frac{ix\mathfrak {I}\mu _j}{\lambda _j} \right) \right] \nonumber \\&\quad e^{\frac{-x}{\lambda _j}} \mathrm{d}x = c T. \end{aligned}$$

(9)

Now, using Lemma 3a, (8) and (9), inequality (7) gives an inequality

$$\begin{aligned}&\frac{16}{\pi ^2} R T \ge \frac{16 m_F T^3}{(4 \pi ^2 + T^2)^2} e^{\frac{T}{2}} - 8 K_F e^{\frac{T}{2}(2 \theta + 1)} - 2 r_2(T) \\&\quad + \log q_F - 2 \sum _{j=1}^{r}\lambda _j \log (2 \pi \lambda _j) + cT. \end{aligned}$$

For T large enough there exists $\delta >0$ such that

$$\begin{aligned} \Bigg |\frac{16 m_F T^3}{(4 \pi ^2 + T^2)^2} e^{\frac{T}{2}} - 2 r_2(T) - 2 \sum _{j=1}^{r}\lambda _j \log (2 \pi \lambda _j) + cT \Bigg | \le 8 \delta e^{\frac{T}{2}(2 \theta + 1)}, \end{aligned}$$

hence

$$\begin{aligned} \frac{16}{\pi ^2} R T \ge - 8 (K_F + \delta ) e^{\frac{T}{2}(2 \theta + 1)} + \log q_F. \end{aligned}$$

Setting

$$\begin{aligned} A=\frac{8}{\pi ^2} \frac{(4m_F + 1)\log q_F + B(F)}{ \log \log q_F}, \, B = 8(K_F + \delta ), \, C=\log q_F \, \, \, \text {and} \,\,\, \alpha = \theta + \frac{1}{2} \end{aligned}$$

the result of Theorem 2 easily follows from Theorem 1 and Lemma 4.

5.3 Proof of Theorem 3

For $T = \sqrt{2} \pi / h$, applying the explicit formula (3) as in the proof of Theorem 2 we obtain the inequality (7).

For $a_F(n)\ge 0$, by  [21, p. 294] we have $m_F > 0$, hence using (8), (9) and Lemma 3b, inequality (7) yields the inequality

$$\begin{aligned} \frac{16}{\pi ^2} R T \ge \frac{16 m_F T^3}{(4 \pi ^2 + T^2)^2} e^{\frac{T}{2}} - 8 m_F e^{\frac{T}{2}} - 2 P_F(T) + \log q_F - 2 \sum _{j=1}^{r}\lambda _j \log (2 \pi \lambda _j) + cT. \end{aligned}$$

For T large enough there exists $\tau >0$ such that

$$\begin{aligned} \Bigg |\frac{16 m_F T^3}{(4 \pi ^2 + T^2)^2} e^{\frac{T}{2}} - 2 P_F(T) - 2 \sum _{j=1}^{r}\lambda _j \log (2 \pi \lambda _j) + cT \Bigg | \le 8 \tau e^{\frac{T}{2}}, \end{aligned}$$

hence

$$\begin{aligned} \frac{16}{\pi ^2} R T \ge - 8 (m_F + \tau ) e^{\frac{T}{2}} + \log q_F. \end{aligned}$$

Setting

$$\begin{aligned} A=\frac{8}{\pi ^2} \frac{(4m_F + 1)\log q_F + B(F)}{ \log \log q_F}, \, B = 8 (m_F + \tau ), \, C=\log q_F \, \, \, \text {and} \,\,\, \alpha = \frac{1}{2} \end{aligned}$$

the result of Theorem 3 easily follows from Theorem 1 and Lemma 4.

6 An Application to Automorphic L-Functions

Let $\pi$ be an irreducible unitary cuspidal representation of $GL_N(\mathbb {Q})$. Then the (finite) automorphic L-function $L(s, \pi )$ attached to $\pi$ is given by products of local factors for $\mathfrak {R}s > 1$ (see e.g.  [4])

$$\begin{aligned} L(s, \pi ) = \prod _{p} \prod _{j=1}^{N} \big ( 1 - \alpha _{p, j} (\pi ) p^{-s} \big )^{-1}=\sum _{n=1}^{\infty } \frac{a_n(\pi )}{n^s}, \end{aligned}$$

where

$$\begin{aligned} a_n(\pi ) = \sum _{j=1}^{N} \alpha _{p, j} (\pi )^{k}. \end{aligned}$$

Therefore

$$\begin{aligned} \log L(s, \pi ) = \sum _{n=1}^{\infty } \frac{b_n(\pi )}{n^s} \end{aligned}$$

and

$$\begin{aligned} \frac{L'}{L}(s, \pi ) = -\sum _{n=1}^{\infty } \frac{c_n(\pi )}{n^s}, \end{aligned}$$

where

$$\begin{aligned} b_n(\pi ) = \frac{{\varLambda }(n) a_n(\pi )}{\log n}, \end{aligned}$$

$$\begin{aligned} c_n(\pi ) = b_n(\pi )\log n. \end{aligned}$$

(10)

In the series of papers  [16–19], Shahidi has shown that the complete L-function

$$\begin{aligned} {\varLambda }(s, \pi ) = Q(\pi )^{s/2} L_{\infty }(s, \pi _{\infty }) L(s, \pi ), \end{aligned}$$

where $Q(\pi ) > 0$ is the conductor of $\pi$ and

$$\begin{aligned} L_{\infty }(s, \pi _{\infty }) = \prod _{j=1}^{N} {\varGamma }_{\mathbb {R}} (s + \kappa _j (\pi )). \end{aligned}$$

is the archimedean factor, satisfies the functional equation

$$\begin{aligned} {\varLambda }(s, \pi ) = \epsilon (\pi )\overline{{\varLambda }(1 - \bar{s}, \pi )} \end{aligned}$$

with a constant $\epsilon (\pi )$ of absolute value 1. Here, ${\varGamma }_{\mathbb {R}} (s) = \pi ^{-s/2} {\varGamma }(s/2)$ and the parameters $\kappa _j$ satisfy the inequality $\mathfrak {R}\kappa _j > -1/2$, as proved by Rudnick and Sarnak in  [14]. Jacquet and Shalika in  [5] proved that

$$\begin{aligned} L(s, \pi ) \ne 0 \quad \text {for} \quad \mathfrak {R}s =1. \end{aligned}$$

It is easy to see that $L(s, \pi ) \in \mathcal {S}^{\sharp \flat }$  [10, pp. 533–534] with $r=N, Q_F = Q(\pi )^{1/2} \pi ^{-N/2}, \lambda _j = \frac{1}{2}, \mu _j = \frac{1}{2} \kappa _j(\pi ), j = 1, \ldots , N, d_F = N$ and the parameters $\kappa _j$ satisfy the inequality $\mathfrak {R}\kappa _j > - 1/2.$

Assuming GRH for automorphic L-functions and applying results of Theorems 1 and 2 to $L(s, \pi ) \in \mathcal {S}^{\sharp \flat }$ we get the following corollaries.

Corollary 3

Let R be the multiplicity of the eventual zero at the central point 1 / 2 of $L(s, \pi )$ such that $\mathfrak {R}({c_n} (\pi )) \ge 0$ and let

$$\begin{aligned} B(L) = \sum _{j=1}^{N} \mathfrak {R}\Big ( {\varPsi }\Big ( \frac{1}{4} + \frac{1}{2}\kappa _j(\pi ) \Big ) \Big ) - N \log \pi . \end{aligned}$$

• (a) If $Q(\pi ) > e$ then

  $$\begin{aligned} R \le \frac{(4m_L + 1)\log Q(\pi ) + B(L)}{2 \log \log Q(\pi )} . \end{aligned}$$

• (b) If $0 < Q(\pi ) \le e$ then

  • (i) $R=0$, when $N>1$ or $N=1$ and $\pi \ne Id$,

  • (ii) $R \le \frac{4m_L e^{ W \big ( \frac{1 - \gamma - \pi /2 - \log 8\pi }{4 e }\big ) + 1} + 1 - \gamma - \pi /2 - \log 8\pi }{2 \Big ( W \Big ( \frac{1 - \gamma - \pi /2 - \log 8\pi }{4 e }\Big ) + 1\Big )}.$

where W denotes the Lambert function. Specially, if $L(s, \pi ) \ne \zeta (s)$ is automorphic L-function with analytic conductor $Q(\pi )$ less than or equal to e, then $L(s, \pi )$ is non-vanishing at central point $s=1/2$.

Proof

Part a) of the statement follows immediately from Theorem 1a with $r=N$ and $\lambda _j = \frac{1}{2}, \mu _j = \frac{1}{2} \kappa _j(\pi ), j = 1, \ldots , N, Q_F = Q(\pi )^{1/2} \pi ^{-N/2}$.

When $N>1$ or $N=1$ and $\pi \ne Id L$-function is entire, hence $m_L = 0$, thus Theorem 1b yields $R=0$.

When $N=1$ and $\pi = Id, L(s, \pi ) = \zeta (s)$, hence $B(L) = {\varPsi }(1/4) - \log \pi$. Moreover, ${\varPsi }(1/4) = -\frac{\pi }{2} - 3\log 2 - \gamma$, thus $B(L) = -\frac{\pi }{2} - \log 8\pi - \gamma$. The proof is complete. $\square$

Corollary 4

Let h be the height of the first zero with imaginary part different from zero of the function $L(s, \pi )$. Assume that $L(s, \pi )$ satisfies axiom (5) of the Selberg class and $\mathfrak {R}({c_n} (\pi )) \ge 0$, where $c_n(\pi )$ are given by (10). Then, for $Q(\pi ) > e$ we have the bound

$$\begin{aligned} h \le \max \Bigg \{ \frac{16 \sqrt{2}\Big [\log Q(\pi ) + B(L) \Big ]}{\pi \log Q(\pi ) \log \log Q(\pi )}, \frac{(2 \theta + 1) \pi }{\sqrt{2} \log [ \log Q(\pi ) / 16(K_L+ \delta ) ]} \Bigg \}. \end{aligned}$$

Here $m_L$ is defined in axiom (2) of the Selberg class, B(L) is given in Corollary 3, $K_L$ is defined in Lemma 3, $\theta < 1/2$ and $\delta > 0$.

Proof

We proceed analogously as in the proof of Corollary 3, by putting $r=N, Q_F = Q(\pi )^{\frac{1}{2}} \pi ^{-\frac{N}{2}}, \lambda _j = \frac{1}{2}, \mu _j = \frac{1}{2} \kappa _j(\pi ), j = 1, \ldots , N$. Then, we observe that the conductor $q_{\zeta }$ of the Riemann zeta function is equal to one, hence applying the relation (1) with $r=1, \quad \lambda _1 =\frac{1}{2}$, we see that $q_{\zeta } = 1 = \pi Q_{\zeta }^2$, thus the analytic conductor of the Riemann zeta function is also equal to 1.

Therefore, assumption $Q(\pi ) > e$ yields that $L(s, \pi ) \ne \zeta (s)$, hence $L(s, \pi )$ is holomorphic and $m_L =0$.

Once we observe that,

$$\begin{aligned} q_L = (2\pi )^N Q_L^2 \Big (\frac{1}{2} \Big )^N = \pi ^N Q(\pi ) \pi ^{-N} = Q(\pi ), \end{aligned}$$

the corollary follows immediately from Theorem 2. $\square$