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Let $X$ be a real prehomogeneous vector space under a reductive group $G$,such that $X$ is an absolutely spherical $G$-variety with affine open orbit. Wedefine local zeta integrals that involve the integration of Schwartz-Bruhatfunctions on $X$ against generalized matrix coefficients of admissiblerepresentations of $G(\mathbb{R})$, twisted by complex powers of relativeinvariants. We establish the convergence of these integrals in some range, themeromorphic continuation as well as a functional equation in terms of abstract$\gamma$-factors. This subsumes the Archimedean zeta integrals ofGodement-Jacquet, those of Sato-Shintani (in the spherical case), and theprevious works of Bopp-Rubenthaler. The proof of functional equations is basedon Knop's results on Capelli operators.

Generalized zeta integrals on certain real prehomogeneous vector spaces