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# $L$-Functions for $\GSp(2)\times \GL(2)$: Archimedean Theory and Applications

Published:2009-04-01
Printed: Apr 2009
• Tomonori Moriyama
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## Abstract

Let $\Pi$ be a generic cuspidal automorphic representation of $\GSp(2)$ defined over a totally real algebraic number field $\gfk$ whose archimedean type is either a (limit of) large discrete series representation or a certain principal series representation. Through explicit computation of archimedean local zeta integrals, we prove the functional equation of tensor product $L$-functions $L(s,\Pi \times \sigma)$ for an arbitrary cuspidal automorphic representation $\sigma$ of $\GL(2)$. We also give an application to the spinor $L$-function of $\Pi$.
 MSC Classifications: 11F70 - Representation-theoretic methods; automorphic representations over local and global fields 11F41 - Automorphic forms on ${\rm GL}(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] 11F46 - Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms