We investigate the analytic properties of a Dirichlet series involving the Fourier–Jacobi coefficients of two cusp forms for orthogonal groups of signature \((2,n+2)\) . Using an orthogonal Eisenstein series of Klingen type, we obtain an integral representation for this Dirichlet series. In the case when the corresponding lattice has only one 1-dimensional cusp, we rewrite this Eisenstein series in the form of an Epstein zeta function. If additionally \(4 \mid n\) , we deduce a theta correspondence between this Eisenstein series and a Siegel Eisenstein series for the symplectic group of degree 2. We obtain, in this way, the meromorphic continuation of the Dirichlet series to \(\mathbb {C}\) as a corollary. In the case of the \(E_8\) lattice, we are able to further deduce a precise functional equation for the Dirichlet series.
