$L(\sigma,\chi)$ is real for real $\sigma$ when $\chi$ is a real character

Reality of $L(s,\chi)$ on the real axis for real characters. Let $\chi$ be a real (quadratic: all values in $\{0, 1, -1\}$) non-principal Dirichlet character modulo $q \ge 1$. Then for every real $\sigma$, the value $L(\sigma,\chi)$ of the entire function $L(s,\chi)$ is a real number:

For $\sigma > 1$ this is clear from the Dirichlet series $\sum \chi(n)n^{-\sigma}$, whose terms are real; for the remaining $\sigma$ it follows by analytic continuation (for instance from the Taylor expansion of the entire function $L(s,\chi)$ about $s = 2$, all of whose coefficients are real). It is the fact that lets one speak of real zeros $\beta_1$ of $L(s,\chi)$ and of the sign of $L(\sigma,\chi)$ on $[1-\varepsilon, 1]$ in Siegel's theorem.

Formalization Note. The non-principality hypothesis is included so that $L(s,\chi)$ is entire; for the principal character the value assigned at the pole $s = 1$ is not covered by this statement.