On analytic properties of the standard zeta function attached to a vector-valued modular form (Q2093673)

The main objective of the present paper is to examine the analytic properties of a certain \(L\)-function of a vector-valued modular form of type \(\rho_{L,1}\) which is a finite dimensional representation of the symplectic modular group \(\mathrm{SL}(2,\mathbb{Z})\). The main tool in this regard is to prove the Garret-Böcherer decomposition of a non-holomorphic vector-valued Siegel Eisenstein series \(E^2_{l,0}\). As consequence, it is shown that the standard zeta function associated to a vector-valued common eigenform \(f\) for the Weil representation \(\rho_{L,1}\) can be meromorphically continued to the whole complex plane and that it satisfies a functional equation.