M. Peter and Ueno introduced two variables Dirichlet series accociated with non-degnerate quadratic forms, investigated their analytic properties and constructed holomorphic modular forms using the converse theorem as an application. This has been extended so that it includes Maass forms by an modular-distribution approach by Sato-Ueno-Tamura-Miyazaki-Sugiyama. On the other hand Duke-Imamoglu has shown that a certain series coming from CM point average of SL_2(Z)-non-holomorphic Eisenstein series is a series of Fourier coefficients of a Maass form. This is a Katok-Sarnak correspondence for Eisenstein series. Keeping this in mind we would like to state the following. Fourier coefficients and the Mellin transform of non-holomorphic Eisenstein series of matrix coefficients can be interpreted as the Melloin transform of the above two variable Dirichlet series. Therefore, the above modular form/Maass form is a modular form/Maass form naturally associated with Jacobi-Eisenstein series of matrix coefficients. In particular, as an extension of Duke-Imamoglu, the series made by CM points average of non-holomorphic Eisenstein seris of hyperbolic space of higher dimension is a series of the Fourier coefficients of this Maass form (but the non-degenerate quadratic form has to be positive-definite).