Mathematics > Number Theory
[Submitted on 1 May 2018 (v1), last revised 23 Jul 2021 (this version, v4)]
Title:Quantitative non-vanishing of central values of certain $L$-functions on ${\rm GL}(2)\times {\rm GL}(3)$
View PDFAbstract:Let $\phi$ be an even Hecke-Maass cusp form on ${\rm SL}_2(\mathbb{Z})$ whose $L$-function does not vanish at the center of the functional equation. In this article, we obtain an exact formula of the average of triple products of $\phi$, $f$ and $\bar f$, where $f$ runs over an orthonormal basis $H_k$ of Hecke eigen elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$ of a fixed weight $k\geq 4$. As an application, we prove a quantitative non-vanishing results on the central values for the family of degree $6$ $L$-functions $L(s,\phi \times {\rm Ad}\,f)$ with $f$ in the union of $H_k$ $({\rm K} \leq k < 2{\rm K})$ as ${\rm K}\rightarrow \infty$.
Submission history
From: Shingo Sugiyama [view email][v1] Tue, 1 May 2018 06:47:41 UTC (30 KB)
[v2] Wed, 6 Mar 2019 11:57:22 UTC (36 KB)
[v3] Wed, 13 Jan 2021 02:54:37 UTC (37 KB)
[v4] Fri, 23 Jul 2021 13:19:33 UTC (38 KB)
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