Sifting for small split primes of an imaginary quadratic field in a given ideal class

Let D>3, D≡3(mod4) be a prime, and let C be an ideal class in the field Q(−D−−−√). We give a new proof that p(D,C), the smallest norm of a split prime p∈C, satisfies p(D,C)≪DL for some absolute constant L. Our proof is sieve-theoretic. In particular, this allows us to avoid the use of log-free zero-density estimates (for class group L-functions) and the repulsion properties of exceptional zeros, two crucial inputs to previous proofs of this result.