According to [Lemke-Oliver][1], irreducible quadratic polynomials (with positive leading coefficient and with $\rho(2)<2$) represent $p_1p_2$ infinitely often, with $p_1,p_2$ two distinct primes.

Is there anything known about the distribution of $(p_1/p_2)$, with $(\cdot,\cdot)$ the Legendre symbol? I am working with a specific family of polynomials which appear to have $(p_1/p_2)=\pm1$, both options infinitely often (and with the same density). Is there any known result along these lines?


  [1]: http://math.tufts.edu/faculty/rlemkeoliver/papers/04-Quadratic.pdf