  We show that the sum of the traces of Frobenius elements of Artin
$L$-functions in a family of $G$-fields satisfies the Gaussian distribution
under certain counting conjectures. We prove the counting conjectures for $S_4$
and $S_5$-fields. We also show central limit theorem for modular form
$L$-functions with the trivial central character with respect to congruence
subgroups as the level goes to infinity.
