Is true that $Z(G)/N = Z(G/N)$ for connected topological groups?

Question

Let $G$ be a connected topological group and $N$ a discrete normal subgroup of $G$.

Is it true that $Z(G)/N = Z(G/N)$, where $Z(G)$ denotes the center of $G$?

I know that every discrete normal subgroup of $G$ is contained in its center.