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Proceedings of the American Mathematical Society

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Non-vanishing of derivatives of $GL(3) \times GL(2)$ $L$-functions


Author: Qingfeng Sun
Journal: Proc. Amer. Math. Soc. 140 (2012), 449-463
MSC (2010): Primary 11F67, 11F12, 11F30
DOI: https://doi.org/10.1090/S0002-9939-2011-10947-X
Published electronically: June 14, 2011
MathSciNet review: 2846314
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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 $Lโ€™(1/2,f\times g)\neq 0$.


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Additional Information

Qingfeng Sun
Affiliation: School of Mathematics and Statistics, Shandong University at Weihai, Weihai 264209, Peopleโ€™s Republic of China
Email: qfsun@mail.sdu.edu.cn

Keywords: Non-vanishing, $GL(3) \times GL(2)$ $L$-functions
Received by editor(s): April 2, 2010
Received by editor(s) in revised form: September 24, 2010, November 28, 2010, and November 29, 2010
Published electronically: June 14, 2011
Additional Notes: The author was supported by National Natural Science Foundation of China (grant No.ย 10971119).
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.