Sum of irreducible character values in a row of the character table

Question

If $\chi$ is a nontrivial irreducible character of $G$ (a finite group), define $S_{\chi}:= \sum_{x \in G} \chi(x)$. In terms of conjugacy classes $\mathcal{C}$, this is $\sum_{\mathcal{C}} |\mathcal{C}| \chi(\mathcal{C})$. Is there a nice condition that guarantees $S_{\chi}=0$?

I've noticed that this occurs, for instance, with $S_5$. I'd love a description of this phenomenon and a proof, if possible.

Answer 1

As @ah11950 states in the comments, $$\sum_{g\in G} \chi(g) = |G|\langle \chi, \mathbb 1\rangle,$$ so $S_{\chi} = 0$ if and only if $1$ doesn't appear as a direct summand of $\chi$.