A theorem of M¨uger asserts that the center Z(C) of a spherical fusion category C over k is a modular fusion category if k is an algebraically closed field and the dimension of C is invertible. We generalize this result to the case where k is an arbitrary commutative ring, without restriction on the dimension of the category. Moreover we construct a variant of the Reshetikhin- Turaev invariant associated to Z(C), still defined when dimC is not invertible, and give an algorithm for computing this invariant in terms of certain explicit morphisms in the category C. Our approach is based on (a) Lyubashenko's construction of the Reshetikhin-Turaev invariant in terms of the coend of a ribbon category; (b) an explicit algorithm for computing this invariant via Hopf diagrams; (c) an algebraic interpretation of the center of C as the category of modules over a canonical Hopf monad on C; (d) a generalization of the Drinfeld double construction to Hopf monads which, applied to the canonical Hopf monad of C, provides an explicit description of the coend of Z(C) in terms of the category C.