Ẑ invariants at rational τ

Ẑ invariants of 3-manifolds were introduced as series in q = e2πiτ in order to categorify Witten-Reshetikhin-Turaev invariants corresponding to τ = 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 Ẑ invariants for τ ∈ ℚ. We present the reasoning leading to this conjecture and test it on various 3-manifolds.