Effective moments of Dirichlet $L$-functions in Galois orbits

Abstract

Khan, Milićević, and Ngo evaluated the second moment of $L$-functions associated to certain Galois orbits of primitive Dirichlet characters to modulus a large power of any fixed odd prime $p$. Their results depend on $p$-adic Diophantine approximation and are ineffective, in the sense of computability. We obtain an effective asymptotic for this second moment in the case of $p=3,5,7$.