A theorem of Müger asserts that the center $\mathcal{Z}(\mathcal{C})$ of a spherical $\mathbb{k}$-linear category $\mathcal{C}$ is a modular category if $\mathbb{k}$ is an algebraically closed field and the dimension of $\mathcal{C}$ is invertible. We generalize this result to the case where $\mathbb{k}$ is an arbitrary commutative ring, without restriction on the dimension of the category. Moreover we construct the analogue of the Reshetikhin-Turaev invariant associated to $\mathcal{Z}(\mathcal{C})$ and give an algorithm for computing this invariant in terms of certain explicit morphisms in the category $\mathcal{C}$. 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 $\mathcal{C}$ as the category of modules over a certain Hopf monad Z on the category $\mathcal{C}$; (d) a generalization of the classical notion of Drinfeld double to Hopf monads, which, applied to the Hopf monad Z, provides an explicit description of the coend of $\mathcal{Z}(\mathcal{C})$ in terms of the category $\mathcal{C}$.