Let $G$ be a group locally isomorphic to $SL(2,\mathbb R)$, and with finite center, then $G$ is a covering space of $PSL(2,\mathbb R)$

Question

I'm reading from a book about lie groups and representations. The author mentions without prove, what should be trivial, that:

If $G$ is a group locally isomorphic to $SL(2,\mathbb R)$ and $G$ has finite center, then there is a finite sheet covering $f:\:G\to PSL(2,\mathbb R)$ such that $Z_G=\ker f$.

I can't see why this is true. My attempt was trying to use the adjoint group $Ad(G)$ and its relation to the inner automorphism group (which for $SL(2,\mathbb R)$ I know it to be $PSL(2,\mathbb R)$ , to show that the adjoint representation would satisfy the conditions.