Non-vanishing of quadratic twists of modular $L$-functions of prime-related moduli

TL;DR

This paper proves, under the assumption of the generalized Riemann hypothesis, that a positive proportion of quadratic twists of certain modular L-functions do not vanish at the center, focusing on moduli related to prime numbers.

Contribution

It establishes a positive proportion non-vanishing result for quadratic twists of modular L-functions with moduli of the form 8p, where p is an odd prime, assuming GRH.

Findings

Positive proportion of non-vanishing L-values under GRH

Focus on quadratic twists with moduli 8p

Results applicable to primes p

Abstract

In this paper, we study central values of the family of quadratic twists of modular $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish a positive proportion non-vanishing result for the corresponding $L$-values.