Dirichlet L-function (reviewed)

A Dirichlet L-function is an L-function defined by a Dirichlet series of the form ${\displaystyle L(s, \chi)=\sum_{n=1}^\infty \frac {\chi(n)}{n^s}}$, where $\chi$ is a Dirichlet character.

Dirichlet L-functions can be meromorphically continued to the complex plane, have an Euler product ${\displaystyle L(s, \chi)= \prod_p (1 - \chi(p) p^{-s} )^{-1}}$, and satisfy a functional equation of the form \[ \Lambda(s,\chi) = q^{s/2} \Gamma_{\mathbb R} (s) L(s,\chi) = \varepsilon_\chi \overline{\Lambda}(1-s), \] where $q$ is the conductor of $\chi$.

These L-functions were introduced by Peter Gustav Lejeune Dirichlet in the mid-1800s as a tool to study prime numbers in arithmetic progressions.