  We compute the number of irreducible linear representations of self-similar
branch groups, by expressing these numbers as the co\"efficients a_n of a
Dirichlet series sum a_n n^{-s}.
  We show that this Dirichlet series has a positive abscissa of convergence, is
algebraic over the ring Q[2^{-s},...,P^{-s}] for some integer P, and show that
it can be analytically continued (through root singularities) to the left
half-plane.
  We compute the abscissa of convergence and the functional equation for some
prominent examples of branch groups, such as the Grigorchuk and Gupta-Sidki
groups.
