Quantitative non-vanishing of central values of certain $L$-functions on $\mathrm{GL}(2)\times \mathrm{GL}(3)$

Abstract

Let $\varphi$ be an even Hecke-Maass cusp form on ${\mathrm{SL}}_2(\mathbb{Z})$ 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 $\varphi$, $f$ and $\bar{f}$, where $f$ runs over an orthonormal basis $H_k$ of Hecke eigen elliptic cusp forms on ${\mathrm{SL}}_2(\mathbb{Z})$ of a fixed weight $k\ge 4$. As an application, we prove a quantitative non-vanishing results on the central values for the family of degree $6$ $L$-functions $L(s,\varphi \times \mathrm{Ad}f)$ with $f$ in the union of $H_k$ $(\mathrm{K}\le k<2\mathrm{K})$ as $\mathrm{K}\to \infty$.

Some typos are fixed. Theorem 1.4, Lemma 3.2 and Proposition 3.3 are corrected

References in corpus (1)

• An explicit trace formula of Jacquet-Zagier type for Hilbert modular forms

Cited by in corpus (1)

• Wide moments of $L$-functions I: Twists by class group characters of imaginary quadratic fields