Enzymatic replication in the absence of compartmentalization has often been rejected as plausible hypothesis for the origin of life, because unspecific replication does not give a selective advantage for functional enzymes. We analyze a quasispecies model for specifically replicating enzymes, where replication rates decrease (self-specific) or increase (cross-specific) with Hamming distance between enzyme and substrate. We combine stochastic simulations, numerical solutions of the rate equations and analytical solutions of an excellent approximation, which allows to compute the whole population distribution and the full phase diagram. This yields conditions for localization about a master sequence, and, in the case of cross-specificity, coexistence of complementary populations, in dependence on mutation rate, selection strength and specificity degree.