Central character of an L-function (reviewed)

An L-function has an Euler product of the form $L(s) = \prod_p L_p(p^{-s})^{-1}$ where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called central character of the L-function. Here, $N$ is the conductor of $L$.