A Fourier-Jacobi Dirichlet series for cusp forms on orthogonal groups

We investigate a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms F, G for orthogonal groups of signature $(2,n+2)$ . In the case when F is a Hecke eigenform and G is a Maass lift of a Poincaré series, we establish a connection with the standard L-function attached to F. What is more, we find explicit choices of orthogonal groups, for which we obtain a clear-cut Euler product expression for this Dirichlet series. Through our considerations, we recover a classical result for Siegel modular forms, first introduced by Kohnen and Skoruppa, but also provide a range of new examples, which can be related to other kinds of modular forms, such as paramodular, Hermitian, and quaternionic.