Nonvanishing of Dirichlet L-functions, II

Nonvanishing of quadratic Dirichlet L - functions at s

The Generalized Riemann Hypothesis (GRH) states that all non-trivial zeros of Dirichlet L-functions lie on the line Re(s) = 12 . Further, it is believed that there are no Q-linear relations among the non-negative ordinates of these zeros. In particular, it is expected that L( 1 2 , χ) 6= 0 for all primitive characters χ, but this remains still unproved. This appears to have been first conjectur...

On Artin’s L-functions. II: Dirichlet Coefficients

Nonvanishing of certain Rankin-Selberg L-functions

In this article we prove that given a holomorphic cusp form f and any point s0 in the complex plane, there is a holomorphic cusp form g such that the Rankin-Selberg L-function L(s, f × g) is non-zero at s0. Résumé: Dans cet article, on prouve le résultat suivant. Etat donné une forme holomorphe cuspidale f et un point quelquonque du plan complexe, il existe une forme holomorphe cuspidale g tell...

Effective Nonvanishing of Canonical Hecke L-functions

Motivated by work of Gross, Rohrlich, and more recently Kim, Masri, and Yang, we investigate the nonvanishing of central values of L-functions of “canonical” weight 2k−1 Hecke characters for Q( √ −p), where 3 < p ≡ 3 (mod 4) is prime. Using the work of Rodriguez-Villegas and Zagier, we show that there are nonvanishing central values provided that p ≥ 6.5(k−1) and (−1) ( 2 p ) = 1. Moreover, we ...

Nonvanishing of L - functions on < ( s ) = 1

In [Ja-Sh], Jacquet and Shalika use the spectral theory of Eisenstein series to establish a new result concerning the nonvanishing of L-functions on <(s) = 1. Specifically they show that the standard L-function L(s, π) of an automorphic cusp form π on GLm is nonzero for <(s) = 1. We analyze this method, make it effective and also compare it with the more standard methods. This note is based on ...