Primes represented by quadratic forms and the Weil abscissa of abelian profinite groups

Here we show that the Weil abscissa of the procyclic groups ${\prod }_{p\in S}{\mathbb{Z}}_p$ equals $2$ for three sets $S$: (i) the set of primes $p\equiv 13$, (ii) the set of primes $p\equiv 14$ and (iii) the set of primes $p\equiv 1,38$. Our argument is based on the observation that integers all of whose prime factors lie in $S$ can be represented by a suitable binary quadratic form, which allows us to use a theorem of Iwaniec to exhibit a minorant for the Weil representation zeta function.