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# Values of Twisted Tensor $L$-functions of Automorphic Forms Over Imaginary Quadratic Fields

Published:2014-03-18
Printed: Oct 2014
• Dominic Lanphier,
Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101
• Howard Skogman,
Department of Mathematics, SUNY Brockport, Brockport, NY 14420
 Format: LaTeX MathJax PDF

## Abstract

Let $K$ be a complex quadratic extension of $\mathbb{Q}$ and let $\mathbb{A}_K$ denote the adeles of $K$. We find special values at all of the critical points of twisted tensor $L$-functions attached to cohomological cuspforms on $GL_2(\mathbb{A}_K)$, and establish Galois equivariance of the values. To investigate the values, we determine the archimedean factors of a class of integral representations of these $L$-functions, thus proving a conjecture due to Ghate. We also investigate analytic properties of these $L$-functions, such as their functional equations.
 Keywords: twisted tensor $L$-function, cuspform, hypergeometric series
 MSC Classifications: 11F67 - Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F37 - Forms of half-integer weight; nonholomorphic modular forms

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