  We compute the superconformal index of 3d $\mathcal{N}=2$ superconformal
theories obtained from $N$ M5-branes wrapped on a hyperbolic 3-manifold.
Exploiting the 3d-3d correspondence, we use perturbative invariants of
$SL(N,\mathbb{C})$ Chern-Simons theory to determine the superconformal index in
the large $N$ limit, including corrections logarithmic in $N$. The leading
order partition function provides a microscopic foundation for the entropy
function of the dual rotating asymptotically AdS$_4$ black holes. We also
verify that the supergravity one-loop contribution to the $\log N$ term
coincides with the field theoretic result. We propose a 3d-3d formulation for
the refined topologically twisted index, and provide strong evidence in support
of its vanishing --- which agrees with the fact that the expected dual rotating
magnetically-charged black hole does not exist. This provides an interesting
link between gravity and a tantalizing mathematical result.
