  In this article we define and study a zeta function $\zeta_G$ - similar to
the Hasse-Weil zeta function - which enumerates absolutely irreducible
representations over finite fields of a (profinite) group $G$. The zeta
function converges on a complex half-plane for all UBERG groups and admits an
Euler product decomposition. Our motivation for this investigation is the
observation that the reciprocal value $\zeta_G(k)^{-1}$ at a positive integer
$k$ coincides with the probability that $k$ random elements generate the
completed group ring of $G$. The explicit formulas obtained so far suggest that
$\zeta_G$ is rather well-behaved.
  A central object of this article is the abscissa of convergence $a(G)$ of
$\zeta_G$. We calculate the abscissae for free abelian, free abelian pro-$p$,
free pro-$p$, free pronilpotent and free prosoluble groups. More generally, we
obtain bounds (and sometimes explicit values) for the abscissae of free
pro-$\mathfrak{C}$ groups, where $\mathfrak{C}$ is a class of finite groups
with prescribed composition factors. We prove that every real number $a \geq 1$
is the abscissa $a(G)$ of some profinite group $G$.
  In addition, we show that the Euler factors of $\zeta_G$ are rational
functions in $p^{-s}$ if $G$ is virtually abelian. For finite groups $G$ we
calculate $\zeta_G$ using the rational representation theory of $G$.
