  In this contribution we announce a complete classification and new exotic
phenomena of the meromorphic structure of $\z$-functions associated to conic
manifolds proved in \cite{KLP1}. In particular, we show that the meromorphic
extensions of these $\z$-functions have, in general, countably many logarithmic
branch cuts on the nonpositive real axis and unusual locations of poles with
arbitrarily large multiplicity. Moreover, we give a precise
algebraic-combinatorial formula to compute the coefficients of the leading
order terms of the singularities.
