Semisimple Lie group with infinite center.

Question

Is there a real semisimple Lie group whose center is infinite?

The center of a Lie group $G$ is the subgroup $$Z_G:=\{g\in G:gh=hg,\forall h\in G\}.$$

If $G$ is semisimple, $\mathrm{Lie}(Z_G)=Z_{\mathfrak{g}}=0$, so $Z_G$ is discrete. But is it possible that $Z_G=\mathbb{Z}$ for example?

Answer 1

For a connected example, take the universal cover of $SL(2,R)$.

Answer 2

Yes, $$\mathrm{SO}(n,\mathbb{R})\times\mathbb{Z}.$$