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

Abstract

Let f f be a fixed self-dual Hecke-Maass cusp form for S L 3 ( Z ) SL_3(\mathbb {Z}) and let B k \mathcal {B}_k be an orthogonal basis of holomorphic cusp forms of weight k ≡ 2 ( m o d 4 ) k \equiv 2(\mathrm {mod} \,4) for S L 2 ( Z ) SL_2(\mathbb {Z}) . We prove an asymptotic formula for the first moment of the first derivative of L ( s , f × g ) L\left (s,f\times g\right ) at the central point s = 1 / 2 s=1/2 , where g g runs over B k \mathcal {B}_k , K ≤ k ≤ 2 K K\leq k\leq 2K , K K large enough. This implies that for each K K large enough there exists g ∈ B k g\in \mathcal {B}_k with K ≤ k ≤ 2 K K\leq k\leq 2K such that L ′ ( 1 / 2 , f × g ) ≠ 0 L’(1/2,f\times g)\neq 0 .