The completed standard $L$-function of modular forms on $G_2$
| Autor: | Çiçek, Fatma, Davidoff, Giuliana, Dijols, Sarah, Hammonds, Trajan, Pollack, Aaron, Roy, Manami |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | The goal of this paper is to provide a complete and refined study of the standard $L$-functions $L(\pi,\operatorname{Std},s)$ for certain non-generic cuspidal automorphic representations $\pi$ of $G_2(\mathbb{A})$. For a cuspidal automorphic representation $\pi$ of $G_2(\mathbb{A})$ that corresponds to a modular form $\varphi$ of level one and of even weight on $G_2$, we explicitly define the completed standard $L$-function, $\Lambda(\pi,\operatorname{Std},s)$. Assuming that a certain Fourier coefficient of $\varphi$ is nonzero, we prove the functional equation $\Lambda(\pi,\operatorname{Std},s) = \Lambda(\pi,\operatorname{Std},1-s)$. Our proof proceeds via a careful analysis of a Rankin-Selberg integral that is due to an earlier work of Gurevich and Segal. Comment: 36 pages |
| Databáze: | arXiv |
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