  We compute the exact one-loop partition function of $\mathbb{Z}_N$ orbifolds
of Euclidean BTZ black hole with the aim to compute the entanglement entropy of
the black hole horizon in string theory as a function of the mass and spin of
the black hole and the $\mathrm{AdS}_3$ radius. We analyze the tachyonic
contribution to the modular integrand for the partition function known for odd
integers $N>1$ and show that it admits an analytic continuation resulting in a
finite answer for the modular integral in the physical region $0< N \leq 1$. We
discuss the flat space limit and the relevance of this computation for quantum
gravity near black hole horizons and holography in relation to the thermal
entropy.
