(Z)over-cap invariants at rational τ

(Z) over cap invariants of 3-manifolds were introduced as series in q = e(2 pi i tau) in order to categorify Witten-Reshetikhin-Turaev invariants corresponding to tau = 1/k. However modularity properties suggest that all roots of unity are on the same footing. The main result of this paper is the expression connecting Reshetikhin-Turaev invariants with (Z) over cap invariants for tau is an element of Q. We present the reasoning leading to this conjecture and test it on various 3-manifolds.