  We study the physics of multiple M5-branes compactified on a hyperbolic
3-manifold. On the one hand, it leads to the 3d-3d correspondence which maps an
$\mathcal{N}=2$ superconformal field theory to a pure Chern-Simons theory on
the 3-manifold. On the other hand, it leads to a warped AdS$_4$ geometry in
M-theory holographically dual to the superconformal field theory. Combining the
holographic duality and the 3d-3d correspondence, we propose a conjecture for
the large $N$ limit of the perturbative free energy of a Chern-Simons theory on
hyperbolic 3-manifold. The conjecture claims that the tree, one-loop and
two-loop terms all share the same $N^3$ scaling behavior and are proportional
to the volume of the 3-manifold, while the three-loop and higher terms are
suppressed at large $N$. Under mild assumptions, we prove the tree and one-loop
parts of the conjecture. For the two-loop part, we test the conjecture
numerically in a number of examples and find precise agreement. We also confirm
the suppression of higher loop terms in a few examples.
