Dirichlet L-function

Remarks

For $\operatorname{Re} s > 1$ there is an Euler product $L(s, \chi) = \prod_p (1 - \chi(p) p^{-s})^{-1}$ over the primes $p$. For the principal character, $L(s, \chi_0) = \zeta(s) \prod_{p \mid q} (1 - p^{-s})$, where $\zeta$ is the Riemann zeta function. Example: for the nonprincipal character $\chi$ modulo $4$, the series at $s = 1$ converges and $L(1, \chi) = 1 - \frac{1}{3} + \frac{1}{5} - \cdots = \frac{\pi}{4}$, by the Leibniz formula for pi. Non-example: $\sum_{n} (-1)^{n+1} n^{-s}$ is a Dirichlet series with $2$-periodic coefficients, but it is not a Dirichlet L-function, since $f(n) = (-1)^{n+1}$ is not a Dirichlet character: $f(4) = -1 \ne 1 = f(2) f(2)$. For a primitive character the completed L-function satisfies a functional equation relating $s$ and $1 - s$, as for the zeta function.