\(L\)-functions for Jacobi forms à la Hecke (Q1340225)

\(L\)-functions for holomorphic Jacobi forms (or more generally automorphic forms on the Jacobi group) have been studied by various authors, e.g. by Gritsenko, Murase and Sugano. Whereas usually attention is focussed on \(L\)-functions with an Euler product decomposition, the present article studies the analytic properties of the suitably defined Mellin transform \(L(f, s)\) of the Jacobi form. By the correspondence between Jacobi forms and modular forms of half-integral weight this Dirichlet series is related to the Mellin transforms of modular forms of half-integral weight, which in general don't have Euler products. Analytic continuation and functional equation are proved for a slightly modified version of the \(L\)-function discussed (which then implies the analytic continuation of \(L(f, s)\) itself).