A note on generalized Bernoulli numbers

In this paper, we consider the zeta function Z ( P, chi, s) associated with a polynomial P ( X) is an element of R[X-1,...,X-r] and chi = (chi (1),...,chi (r)) with chi (j) non- trivial Dirichlet characters, defined by [GRAPHICS] which is absolutely convergent for sufficiently large Re s under some conditions on P ( X). We shall prove that the special value Z ( P,chi,- m) is completely determined by P-m (X) in a simple way. As an immediate application, we give a closed expression for sums of products of any number of generalized Bernoulli numbers.