One-level density of zeros of $\Gamma_1(q)$ $L$-functions
Number Theory
2026-05-21 v2
Abstract
We study the one-level density of zeros for a family of -functions. Assuming GRH, we are able to extend the support of the Fourier transform of the test function to 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 , 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 -functions play a more important role in extending the support than the associated symmetry group.
Cite
@article{arxiv.2512.20066,
title = {One-level density of zeros of $\Gamma_1(q)$ $L$-functions},
author = {Arijit Paul},
journal= {arXiv preprint arXiv:2512.20066},
year = {2026}
}