  Let $\chi$ denote a primitive, non-quadratic Dirichlet character with
conductor $q$, and let $L(s, \chi)$ denote its associated Dirichlet
$L$-function. We show that $|L(1, \chi)| \geq 1/(9.12255 \log(q/\pi))$ for
sufficiently large $q$, and that $|L(1, \chi)| \geq 1/(9.69030 \log(q/\pi))$
for all $q\geq2$, improving some results of Louboutin. The improvements stem
principally from the construction, via simulated annealing, of some real
trigonometric polynomials having particularly favorable properties.
