  In this paper we find lower bounds on higher moments of the error term in the
Chebotarev density theorem. Inspired by the work of Bella\''{\i}che, we
consider general class functions and prove bounds which depend on norms
associated to these functions. Our bounds also involve the ramification and
Galois theoretical information of the underlying extension $L/K$. Under a
natural condition on class functions (which appeared in earlier work), we
obtain that those moments are at least Gaussian. The key tools in our approach
are the application of positivity in the explicit formula followed by
combinatorics on zeros of Artin $L$-functions (which generalize previous work),
as well as precise bounds on Artin conductors.
