Historical question in analytic number theory The analytic continuation and functional equation for the Riemann zeta function were proved in Riemann's 1859 memoir "On the number of primes less than a given magnitude." What is the earliest reference for the analytic continuation and functional equation of Dirichlet L-functions?  Who first proposed that they might satisfy a Riemann hypothesis?  Dirichlet did none of these things; his paper dates from 1837, and as far as I know he only considered his L-functions as functions of a real variable.
 A: 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.
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.  
A: Riemann was the first person who brought complex analysis into the game, but if you ask just about functional equations then he was not the first.  In the 1840s, there were proofs of the functional equation for the $L$-function of the nontrivial character mod 4, relating values at $s$ and $1-s$ for real $s$ between 0 and 1, where the $L$-function is defined by its Dirichlet series. In particular, this happened  before Riemann's work on the zeta-function.  The proofs were due independently to Malmsten and Schlomilch.  Eisenstein had a proof as well (unpublished) which was found in his copy of Gauss' Disquisitiones.  It involves Poisson summation. Eisenstein's proof is dated 1849 and Weil suggested that this might have motivated Riemann in his work on the zeta-function.
For more on Eisenstein's proof, see Weil's "On Eisenstein's Copy of the Disquisitiones" pp. 463--469 of "Alg. Number Theory in honor of K. Iwasawa" Academic Press, Boston, 1989.
A: According to Wikipedia, "an equivalent relationship [equivalent to the functional equation] was conjectured by Euler in 1749". I've seen mention of this in other places too, but of course, that doesn't prove anything.
A: Concerning the statement "An equivalent relationship [equivalent to the functional equation] was conjectured by Euler in 1749". 
This is discussed in Weil's book "Basic number theory." It concerns only the values at integral points: Euler understood $\zeta(1-2k)$ by a simple regularization,
and noticed the relation to $\zeta(2k)$. 
