In this paper, we investigate the zeta function \begin{align*} Z(P,\chi,a,s)&=\sum^\infty_{n_1=1}\cdots\sum^\infty_{n_r=1} \chi_1(n_1)\cdots\chi_r(n_r) \\ &\qquad\cdot P(n_1+a_1,\ldots,n_r+a_r)^{-s}, \end{align*} where $a_i\geq 0$, $\chi_i$ is a Dirichlet character with conductor $N_i$, and $P$ is a polynomial satisfying certain conditions. Its special values at nonpositive integers are closely related to generalized Bernoulli polynomials. Using this fact we can easily get sums of products of Euler polynomials and generalized Bernoulli polynomials.