  We study the moments $M_k(T;\alpha) = \int_T^{2T} |\zeta(s,\alpha)|^{2k}\,dt$
of the Hurwitz zeta function $\zeta(s,\alpha)$ on the critical line, $s = 1/2 +
it$ with a rational shift $\alpha \in \mathbb Q$. We conjecture, in analogy
with the Riemann zeta function, that $M_k(T;\alpha) \sim c_k(\alpha) T (\log
T)^{k^2}$ . Using heuristics from analytic number theory and random matrix
theory, we conjecturally compute $c_k(\alpha)$. In the process, we investigate
moments of products of Dirichlet $L$-functions on the critical line. We prove
our conjectures for the cases $k = 1,2$.
