Editing Dirichlet L-function (section)

== Relation to the Hurwitz zeta function ==
The Dirichlet ''L''-functions may be written as a linear combination of the [[Hurwitz zeta function]] at rational values. Fixing an integer ''k'' ≥ 1, the Dirichlet ''L''-functions for characters modulo ''k'' are linear combinations, with constant coefficients, of the ''ζ''(''s'',''a'') where ''a'' = ''r''/''k'' and ''r'' = 1, 2, ..., ''k''. This means that the Hurwitz zeta function for rational ''a'' has analytic properties that are closely related to the Dirichlet ''L''-functions. Specifically, let ''&chi;'' be a character modulo ''k''.  Then we can write its Dirichlet ''L''-function as:<ref>{{harvnb|Apostol|1976|p=249}}</ref>

:<math>L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}
= \frac{1}{k^s} \sum_{r=1}^k \chi(r) \operatorname{\zeta}\left(s,\frac{r}{k}\right).</math>