Abstract
Let \(F\) be a p-adic field, let \(\psi \) be a non-trivial character of F, and let \(\chi \) be a character of \(F^*\). In this short note we present two new identities involving \(\gamma (s,\chi ,\psi )\), \(\gamma (\psi )\), and \(\widetilde{\gamma }(s,\chi ,\psi )\) along with a duplication formula for \(\gamma (s,\chi ,\psi )\). Here \({\gamma }(s,\chi ,\psi )\) is the Tate \(\gamma \)-factor, \(\gamma (\psi )\) is the Weil index, and \(\widetilde{\gamma }(s,\chi ,\psi )\) is the metaplectic \(\widetilde{\gamma }\)-factor. As a result we give a new proof for a useful identity involving these three factors originally proven by W. Jay Sweet.
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Acknowledgements
We would like to thank Freydoon Shahidi and Fan Gao. The contents of this note emanated from our joint work. We would also like to thank Eyal Kaplan for his valuable comments on the subject matter.
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Szpruch, D. Tate \(\gamma \)-factor, Weil index, and the metaplectic \(\widetilde{\gamma }\) -factor. Ramanujan J 57, 697–706 (2022). https://doi.org/10.1007/s11139-021-00399-7
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DOI: https://doi.org/10.1007/s11139-021-00399-7