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arXiv · 2403.19687

Weighted low-lying zeros of L-functions attached to Siegel modular forms

Shifan Zhao
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Abstract

In this paper, we study weighted low-lying zeros of spinor and standard $L$-functions attached to degree 2 Siegel modular forms. We show the symmetry type of weighted low-lying zeros of spinor $L$-functions is symplectic, for test functions whose Fourier transform have support in $(-1,1)$, extending the previous range $(-\frac{4}{15},\frac{4}{15})$ by E. Kowalski, A. Saha and J. Tsimerman . We then show the symmetry type of weighted low-lying zeros of standard $L$-functions is also symplectic. We further extend the range of support by performing an average over weight. As an application, we discuss non-vanishing of central values of those $L$-functions.

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Shifan Zhao. 2024-03-01. Weighted low-lying zeros of L-functions attached to Siegel modular forms. https://doi.org/10.2140/pjm.2025.335.183

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