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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Non-vanishing of derivatives of $GL(3) \times GL(2)$ $L$-functions
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by Qingfeng Sun PDF
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 $Lโ€™(1/2,f\times g)\neq 0$.
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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
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2846314