Simultaneous non-vanishing of $GL(3)\times GL(2)$ and GL(2) $L$ -functions

preprint2011arXivOpen access

Fix $g$ a Hecke-Maass form for $SL_3(\mathbb{Z})$ . Let $q$ be any large prime number. In the family of holomorphic newforms $f$ of level $q$ and fixed weight, we find the average value of the product $L(\half,g\times f)L(\half,f)$. From this we derive a result on the simultaneous non-vanishing of these $L$ -functions at the central point.

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Simultaneous non-vanishing of $GL(3)\times GL(2)$ and GL(2) $L$ -functions

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Simultaneous non-vanishing of GL(3)×GL(2)GL(3)\times GL(2)GL(3)×GL(2) and GL(2) LLL-functions

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Simultaneous non-vanishing of $GL(3)\times GL(2)$ and GL(2) $L$ -functions