Related papers: A Selberg-type zero-density result for twisted $\r…

We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$ in various aspects. A main tool is Arthur's invariant trace formula. While Shin and Shin-Templier used Euler-Poincar\'e functions at…

We give a full description of the functions $F$ of degree 2 and conductor 1 in the general framework of the extended Selberg class. This is performed by means of a new numerical invariant $\chi_F$, which is easily computed from the data of…

We establish a result of Bombieri-Vinogradov type for the Dirichlet coefficients at prime ideals of the standard $L$-function associated to a self-dual cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n$ over a number field $F$…

We study the range of validity of the density hypothesis for the zeros of $L$-functions associated with cusp Hecke eigenforms $f$ of even integral weight and prove that $N_{f}(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon}$ holds for $\sigma…

It is shown that any number of distinct primitive $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ $L$-functions can simultaneously attain large values on the critical line. This is an unconditional improvement of a general result due to Heap and Li…

After introducing the notion of uniform integrality of critical values of the Rankin-Selberg $L$-functions for $\mathrm{GL}_{n}\times \mathrm{GL}_{n-1}$, we study it when the base field is totally imaginary. For this purpose, we adopt…

In this paper, we estimate the proportion of zeros of Dirichlet $L$-functions on the critical line. Using Feng's mollifier and an asymptotic formula for the mean square of Dirichlet $L$-functions, we prove that averaged over primitive…

Let $q$ be a prime, $\chi$ be a non-principal Dirichlet character $\bmod\ q$ and $L(s,\chi)$ be the associated Dirichlet $L$-function. For every odd prime $q\le 10^7$, we show that $L(1,\chi_\square) > c_{1} \log q$ and $\beta < 1-…

We study the algebraicity of the central critical values of twisted triple product $L$-functions associated to motivic Hilbert cusp forms over a totally real \'etale cubic algebra in the totally unbalanced case. The algebraicity is…

Let f be a Hecke-Maass or holomorphic primitive cusp form for $SL(2,\mathbb{Z})$ with Fourier coefficients $\lambda_{f}(n)$. Let $\chi$ be a primitive Dirichlet character of modulus p, where p is a prime number. In this article we prove the…

We prove the nonvanishing of the twisted central critical values of a class of automorphic $L$-functions for twists by all but finitely many unitary characters in particular infinite families. While this paper focuses on $L$-functions…

Given a nontrivial finite group $G$, we prove the first zero density estimate for families of Dedekind zeta functions associated to Galois extensions $K/\mathbb{Q}$ with $\mathrm{Gal}(K/\mathbb{Q})\cong G$ that does not rely on unproven…

In a previous article we had proved an algebraicity result for the central critical value for L-functions for GL(n) x GL(n-1) over Q assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this…

We obtain explicit formulas for the test vector in the Bessel model and derive the criteria for existence and uniqueness for Bessel models for the unramified, quadratic twists of the Steinberg representation \pi of GSp(4,F), where F is a…

It is proved that \[ \sum_{\chi \bmod q}N(\sigma , T, \chi) \lesssim_{\epsilon} (qT)^{7(1-\sigma)/3+\epsilon}, \] where $N(\sigma, T, \chi)$ denote the number of zeros $\rho = \beta + it$ of $L(s, \chi)$ in the rectangle $\sigma \leq \beta…

We show that if the zeros of an automorphic $L$-function are weighted by the central value of the $L$-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the…

Let $f$ be a Maass form for $SL(3, \mathbb{Z})$ which is fixed and $u_j$ be an orthonormal basis of even Maass forms for $SL(2, \mathbb{Z}),$ we prove an asymptotic formula for the average of the product of the Rankin-Selberg $L$-function…

Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times…

In this paper, we prove that a primitive Hilbert cusp form $\mathbf{g}$ is uniquely determined by the central values of the Rankin-Selberg $L$-functions $L(\mathbf{f}\otimes\mathbf{g}, \frac{1}{2})$, where $\mathbf{f}$ runs through all…

We evaluate asymptotically the smoothed first moment of central values of families of primitive cubic, quartic and sextic Dirichlet $L$-functions, using the method of double Dirichlet series. Quantitative non-vanishing result for these…