One-level densities of large even and odd orthogonal families of automorphic L-functions

TL;DR

This paper establishes one-level density results for automorphic L-functions in orthogonal families, extending the support of test functions and improving non-vanishing bounds under GRH.

Contribution

It provides the strongest known non-vanishing results for these L-functions by extending the support of the Fourier transform of test functions.

Findings

Extended support of Fourier transform to (-3,3)

Achieved strongest non-vanishing results under GRH

Separated treatment of even and odd orthogonal families

Abstract

We prove one-level density results for L-functions attached to primitive forms of level q, averaged over square-free q, conditional on the Generalized Riemann Hypothesis (GRH). We treat the even and odd orthogonal families separately and extend the support of the Fourier transform of the test function to (-3,3). This extended support yields the strongest known non-vanishing results for these families of L-functions and their derivatives at the central point, conditional on GRH.