Dominant zeros of generating functions of primes in arithmetic progressions

For q at least 3 and a coprime to q, let Sq,a be the power series of the primes p at least 3 in the class a modulo q, with exponents p-2, and let Fq,a = 1 - Sq,a. The zeros of least modulus are simple and form the set rq,a times the d-th roots of unity, where d = gcd(q, a-2) and Sq,a(rq,a) = 1. For every R in (0, 1) and every sufficiently large q, none of these functions has a zero in the disk of radius R; for R = 93/100 it is enough to take q at least 39. Thus the disk that contains the three singularities of the unrestricted series does not stay occupied as the modulus grows. For odd q and a = 2, the factorisation Sq,2(z) = H_q(z^q) admits subsequences along which H_q converges to 0 and, for each odd K at least 1, to w^K, uniformly on compact subsets of the unit disk. There is no unique analytic limit, and along these subsequences q times (-log rq,2) tends to 0. Convergence to w + w^3 would imply infinitely many odd moduli for which q+2 and 3q+2 are both prime; that infinitude is a case of Dickson's conjecture and is not proved. An angular signature of the unrestricted series, and the failure of a fixed-modulus asymptotic when q and the scale tend to infinity together, separate these local limits from the zeros of L(s, chi).The record contains the English version (main file), a faithful Portuguese version, and the supplementary archive (Python scripts, figures, and the LaTeX sources). The rational majorant and the two numerical examples are checked by the script verificacao.py. The arguments were not formalised in a proof assistant.Paper 2 of 2 on generating functions built from the odd primes, with exponents of the form p-2.