Unusual poles of the $\zeta$-functions for some regular singular differential operators

We consider the resolvent of a system of first order differential operators with a regular singularity, admitting a family of self-adjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents powers of $\lambda$ which depend on the singularity, and can take even irrational values. The consequences for the pole structure of the corresponding $\zeta$ and $\eta$-functions are also discussed.