Explicit zero-free regions for automorphic $L$-functions
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915513258475520 |
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| author | Creech, Steven Hamieh, Alia Khunger, Simran Sinha, Kaneenika Streipel, Jakob Tsang, Kin Ming |
| author_facet | Creech, Steven Hamieh, Alia Khunger, Simran Sinha, Kaneenika Streipel, Jakob Tsang, Kin Ming |
| contents | Let $L(s,f)$ be the $L$-function associated with a newform $f$ of even weight $k$, squarefree level $N$ and trivial nebentypus. In this paper, we establish a new explicit zero-free region for $L(s,f)$. More precisely, we prove that $L(s,f)$ does not vanish in the region $\Re(s)\geq 1-\frac{1}{C\log(kN\max(1,|\Im(s)|))}$ with $C=16.7053$ if $|\Im(s)|\geq 1$ or $|\Im(s)|\leq \frac{0.30992}{\log(kN)}$ and $C=16.9309$ if $\frac{0.30992}{\log(kN)}<|\Im(s)|\leq 1$. This improves a result of Hoey et al. where $445.994$ was shown to be an admissible value for $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit zero-free regions for automorphic $L$-functions Creech, Steven Hamieh, Alia Khunger, Simran Sinha, Kaneenika Streipel, Jakob Tsang, Kin Ming Number Theory 11M41, 11M26 (primary), 11F11, 11F66, 11F67 (secondary) Let $L(s,f)$ be the $L$-function associated with a newform $f$ of even weight $k$, squarefree level $N$ and trivial nebentypus. In this paper, we establish a new explicit zero-free region for $L(s,f)$. More precisely, we prove that $L(s,f)$ does not vanish in the region $\Re(s)\geq 1-\frac{1}{C\log(kN\max(1,|\Im(s)|))}$ with $C=16.7053$ if $|\Im(s)|\geq 1$ or $|\Im(s)|\leq \frac{0.30992}{\log(kN)}$ and $C=16.9309$ if $\frac{0.30992}{\log(kN)}<|\Im(s)|\leq 1$. This improves a result of Hoey et al. where $445.994$ was shown to be an admissible value for $C$. |
| title | Explicit zero-free regions for automorphic $L$-functions |
| topic | Number Theory 11M41, 11M26 (primary), 11F11, 11F66, 11F67 (secondary) |
| url | https://arxiv.org/abs/2509.20873 |