  We study the pole structure of the $\zeta$-function associated to the
Hamiltonian $H$ of a quantum mechanical particle living in the half-line
$\mathbf{R}^+$, subject to the singular potential $g x^{-2}+x^2$. We show that
$H$ admits nontrivial self-adjoint extensions (SAE) in a given range of values
of the parameter $g$. The $\zeta$-functions of these operators present poles
which depend on $g$ and, in general, do not coincide with half an integer (they
can even be irrational). The corresponding residues depend on the SAE
considered.
