Non-vanishing of derivatives of 𝐺𝐿(3)×𝐺𝐿(2) 𝐿-functions

Sun, Qingfeng

doi:10.1090/S0002-9939-2011-10947-X

Non-vanishing of derivatives of $GL(3) \times GL(2)$ $L$-functions
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Proc. Amer. Math. Soc. 140 (2012), 449-463 Request permission

Abstract:

Let $f$ be a fixed self-dual Hecke-Maass cusp form for $SL_3(\mathbb {Z})$ and let $\mathcal {B}_k$ be an orthogonal basis of holomorphic cusp forms of weight $k \equiv 2(\mathrm {mod} 4)$ for $SL_2(\mathbb {Z})$. We prove an asymptotic formula for the first moment of the first derivative of $L\left (s,f\times g\right )$ at the central point $s=1/2$, where $g$ runs over $\mathcal {B}_k$, $K\leq k\leq 2K$, $K$ large enough. This implies that for each $K$ large enough there exists $g\in \mathcal {B}_k$ with $K\leq k\leq 2K$ such that $L’(1/2,f\times g)\neq 0$.

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