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On zeta functions associated with quadratic forms of variable coefficients

In 1938, C. L. Siegel studied zeta functions of indefinite quadratic forms ([6], c). On the other hand, M. Sato and T. Shintani constructed the general theory of zeta functions of one complex variable associated with prehomogeneous vector spaces in 1974 ([1]). Moreover T. Shintani studied several zeta functions of prehomogeneous vector spaces, especially, “Dirichlet series whose coefficients are class-numbers of integral binary cubic forms” ([3]) and “Zeta functions associated with the vector space of quadratic forms” ([2]).