Davenport (Chapter 9 in Multiplicative Number Theory) claims that the functional equation for Dirichlet L-functions was first given by Hurwitz in 1882 (Werke I, pp.72-88), though only for quadratic characters.  The proof uses what we now call the [Hurwitz zeta function][1].

I was told just yesterday that some people refer to the Riemann Hypothesis for Dirichlet L-functions as the Piltz Hypothesis.  This is confirmed in the [wikipedia article][2].  


  [1]: https://en.wikipedia.org/wiki/Hurwitz_zeta_function
  [2]: https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis