  Motivated by questions of Fouvry and Rudnick on the distribution of Gaussian
primes, we develop a very general setting in which one can study inequities in
the distribution of analogues of primes through analytic properties of
infinitely many $L$-functions. In particular, we give a heuristic argument for
the following claim : for more than half of the prime numbers that can be
written as a sum of two square, the odd square is the square of a positive
integer congruent to $1 \bmod 4$.
