Central limit theorem for Artin L-functions

We show that the sum of the traces of Frobenius elements of Artin (Formula presented.)-functions in a family of (Formula presented.)-fields satisfies the Gaussian distribution under certain counting conjectures. We prove the counting conjectures for (Formula presented.) and (Formula presented.)-fields. We also prove a central limit theorem for the (Formula presented.)-functions of modular forms on congruence subgroups (Formula presented.) as (Formula presented.).ope