  We study splitting densities of primitive elements of a discrete subgroup of
a connected non-compact semisimple Lie group of real rank one with finite
center in another larger such discrete subgroup. When the corresponding cover
of such a locally symmetric negatively curved Riemannian manifold is regular,
the densities can be easily obtained from the results due to Sarnak or Sunada.
Our main interest is a case where the covering is not necessarily regular.
Specifically, for the cases of the modular group and its congruence subgroups,
we determine the splitting densities explicitly. As an application, we study
analytic properties of the zeta function defined by the Euler product over
elements consisting all primitive elements which satisfy a certain splitting
law for a given lifting.
