Twists of automorphic L-functions at the central point

TL;DR

This paper investigates the nonvanishing of twisted automorphic L-functions at the central point, establishing positive proportions of nonvanishing for certain forms and deriving bounds on their average analytic rank.

Contribution

It provides new results on the nonvanishing of twisted automorphic L-functions at the central point and bounds on their average analytic rank for large prime levels.

Findings

Positive proportion of nonvanishing L-functions at s=1/2

Upper bounds on average analytic rank

Analysis of derivatives of L-functions at the center

Abstract

We study the nonvanishing of twists of automorphic L-functions at the centre of the critical strip. Given a primitive character \chi modulo D satisfying some technical conditions, we prove that the twisted L-functions L(f.\chi,s) do not vanish at s=1/2 for a positive proportion of primitive forms of weight 2 and level q, for large prime q. We also investigate the central values of high derivatives of L(f.\chi,s), and from that derive an upper bound for the average analytic rank of the studied L-functions.