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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.20066 |
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| _version_ | 1866914582962896896 |
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| author | Paul, Arijit |
| author_facet | Paul, Arijit |
| contents | We study the one-level density of zeros for a family of $Γ_1(q)$ $L$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20066 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | One-level density of zeros of $Γ_1(q)$ $L$-functions Paul, Arijit Number Theory We study the one-level density of zeros for a family of $Γ_1(q)$ $L$-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. |
| title | One-level density of zeros of $Γ_1(q)$ $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.20066 |