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arXiv · 2512.20066

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

Arijit Paul
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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.

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Arijit Paul. 2025-12-23. One-level density of zeros of $\Gamma_1(q)$ $L$-functions. https://arxiv.org/abs/2512.20066

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