We show that there exists some $\delta > 0$ such that, for any set of integers B with $|B\cap[1,Y]|\gg Y^{1-\delta}$ for all $Y \gg 1$, there are infinitely many primes of the form $a^2+b^2$ with $b\in B$. We prove a quasi-explicit formula for the number of primes of the form $a^2+b^2 \leq X$ with $b \in B$ for any $|B|=X^{1/2-\delta}$ with $\delta < 1/10$ and $B \subseteq [\eta X^{1/2},(1-\eta)X^{1/2}] \cap {\mathbb{Z}}$, in terms of zeros of Hecke L-functions on ${\mathbb{Q}}(i)$. We obtain the expected asymptotic formula for the number of such primes provided that the set B does not have a large subset which consists of multiples of a fixed large integer. In particular, we get an asymptotic formula if B is a sparse subset of primes. For an arbitrary B we obtain a lower bound for the number of primes with a weaker range for $\delta$, by bounding the contribution from potential exceptional characters.

In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2.