On Zeta Functions Associated to Symmetric Matrices, I: An Explicit Form of Zeta Functions

On universality for linear combinations of L-functions

“…For the case n = 2, these zeta functions have been investigated by Siegel, Morita, Shintani, Sato and Arakawa. For the case n ≥ 3, explicit forms of the zeta functions associated to symmetric matrices have been proved by Ibukiyama and Saito in [8] and [9].…”

“…It should be noted that ζ j (s, L) depends only on the determinant η := (−1) n− j and the Hasse invariant θ := (−1) (n− j)(n− j+1)/2 of x ∈ V j n (see [8,Sect. 2]).…”

“…It was also showed in [8,Theorem 1.3] that when n is even, these zeta functions consist of two parts; one part is a product of shifted Riemann zeta functions and the other part is composed of shifted Riemann zeta functions and the Dirichlet series attached to an Eisenstein series of half integral weight of one variable. Moreover, Ibukiyama and Saito [9] gave an explicit form for L-functions associated to the space of n × n symmetric matrices and to Dirichlet characters.…”

On the Density of Unnormalized Tamagawa Numbers of Orthogonal Groups I

“…The principal parts of the global zeta functions for some prehomogeneous vector spaces, including (1.1), were determined by Shintani [25], [26] and Yukie [35], [36], [37]. Ibukiyama-Saito [11] proved an "explicit formula" for the zeta function for (1.1) when the ground field is Q. They expressed the zeta function as a sum of two functions which are products of Riemann zeta functions in the case where n is odd, and expressed the zeta function using Riemann zeta functions and the Eisenstein series of half integral weight in the case where n is even.…”

“…This Z(s) is not the zeta function of the prehomogeneous vector space (1.1). In Part II, we shall express Z(s) as a sum of two Euler products by a technique used in [11], and prove that Z(s) 2 has the rightmost pole at s = is related to the generalized Riemann hypothesis. So it seems difficult to obtain any error term estimate.…”