Infinite product identities for \(L\)-functions (Q819214)

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.