Infinite product identities for \(L\)-functions
type: Item
instance of: scholarly article
title: Infinite product identities for \(L\)-functions
zbMATH Open document ID: 1110.11030
author: K. Joshi
author: Ravi Raghunathan
published in: Illinois Journal of Mathematics
publication date: 2006-03-28T00:00:00Z
review text: Let \[ L(s, \pi \times \chi) = \prod_p \prod_{j=1}^{m_p}(1 - \chi(p) \alpha_j(p)p^{-s})^{-1} \] be Dirichlet series twisted by a character \(\chi\). The authors prove by elementary argument that \[ \prod_{N \geq 1}\;\prod_{\chi \pmod N} L(s+1, \pi \times \chi) = \frac{L(s, \pi)}{L(s+1, \pi)} \] and \[ \prod_{N \geq 1}\;\prod_{\substack{ \chi \pmod N\\ \chi(-1) =1}} L(s+1, \pi \times \chi) = \biggl(\frac{L(s, \pi)}{L_2(s+1, \pi_2)L(s+1, \pi)}\biggr)^{1/2} \] for \(Re(s) > 1 + \delta\). Here \(L_p(s, \pi_p) = \prod_j(1-\alpha_j(p)p^{-j})^{-1}\) denotes the local factor at the prime \(p\). A few examples illustrating special instances of these results are also given.
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 5015537
zbMATH Keywords: Identities for L-functions
reviewed by: Kai-Man Tsang
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5015537