Unusual poles of the ζ-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.