  We give a complete classification and present new exotic phenomena of the
meromorphic structure of $\zeta$-functions associated to general self-adjoint
extensions of Laplace-type operators over conic manifolds. We show that the
meromorphic extensions of these $\zeta$-functions have, in general, countably
many logarithmic branch cuts on the nonpositive real axis and unusual locations
of poles with arbitrarily large multiplicity. The corresponding heat kernel and
resolvent trace expansions also exhibit exotic behaviors with logarithmic terms
of arbitrary positive and negative multiplicity. We also give a precise
algebraic-combinatorial formula to compute the coefficients of the leading
order terms of the singularities.
