One-level density of zeros of $\Gamma_1(q)$ $L$-functions

TL;DR

This paper analyzes the zeros of $ ext{Gamma}_1(q)$ $L$-functions, extending the support of the Fourier transform under GRH, confirming Katz-Sarnak predictions, and establishing a high non-vanishing proportion at the central point.

Contribution

It extends the support of the Fourier transform for the one-level density of $ ext{Gamma}_1(q)$ $L$-functions under GRH and verifies the Katz-Sarnak prediction for this family.

Findings

Support of Fourier transform extended to (-8/3, 8/3)

Proportion of non-vanishing at the central point is at least 62.5%

Highest non-vanishing proportion for a family associated with a unitary group

Abstract

We study the one-level density of zeros for a family of $\Gamma_1(q)$ $L$-functions. Assuming GRH, we are able to extend the support of the Fourier transform of the test function to $\left(-\frac{8}{3},\frac{8}{3}\right)$ and verify the Katz-Sarnak prediction for our unitary family. As an application, we obtain that the proportion of forms in the family with non-vanishing at the central point is at least $62.5\%$, assuming GRH. This is the highest non-vanishing proportion for any family associated with a unitary group. Moreover, this result indicates that the structural properties of $L$-functions play a more important role in extending the support than the associated symmetry group.