A Fourier-Jacobi Dirichlet series attached to modular forms of $SO(2,4)$

We consider a Dirichlet series $D(F,G;s)$ attached to two automorphic forms $F$ and $G$ of an orthogonal group of real signature $(2,4)$, involving their Fourier--Jacobi coefficients. When $F$ is a Hecke eigenform and $G$ a lift of a Jacobi-Poincar\'e series, our main result gives that $D(F,G;s)$ is equal to the standard $L$-function attached to $F$, up to an explicit constant. To establish this, we use a correspondence between binary Hermitian forms and ideals of quaternion algebras, as established by Latimer, together with the fact that the even Clifford algebra of a three-dimensional definite quadratic space can be identified with a quaternion division algebra. Our work should be seen as a generalisation of a work of Kohnen and Skoruppa, whose result corresponds to the case of the orthogonal group of real signature $(2,3)$.