After introducing new zeta functions, he focused efforts on proving their global functional equations for several specific prehomogeneous vector spaces. A special case (the case of spherical Maass cusp forms) of the zeta functions we treat in this paper appears in his paper [@Sato3]. In [@Sato3], the global functional equation is obtained in classical setting (not adelic) and from the point of view of [@KS], the zeta functions are shown to be a Dirichlet series of the Fourier coefficients of certain cusp forms of half integral weight. This fact seems to reflect a hidden relation between the prehomogeneous zeta functions with period integrals and the lifting of automorphic forms. In the course of his proof, he obtained the local functional equation of the local prehomogeneous zeta functions with matrix coefficients of real spherical principal series. The local functional equation of the zeta functions attached to spherical representations are shown in general setting under some conditions by Bopp and Rubenthaler [@BR]. In that paper, they aimed to generalize the Godement-Jacquet theory. Recently, the theory of the local functional equations has greatly developed by the works of Wen-Wei Li [@Li1; @Li2; @Li3; @Li4]. According to his results, the local zeta functions for 