On automorphic L-functions for the Jacobi group of degree one and a relation with L-functions for Jacobi forms

This note discusses an attempt to attach an L-function to an automorphic representation it (with non-trivial central character) of the Jacobi group G(J) of degree one. This group, a semidirect product of SL(2) with a three-dimensional Heisenberg group H, is not reductive and the usual L-function machine (as in Gelbart-Shahidi [GS]) does not work here. So we go back a step and try the definition of local L-factors L(p)(pi s) via certain zeta-integrals zeta(W,s) for elements W from local Whittaker models W-p = W(pi(p), psi(n,r)) which are known to exist here (see [Bell). The classical theory proposes two ways to define L-factors using zeta-integrals.