Let $G$ be a finite group, for each irreducible character $\chi$, we define ${\bf Z}(\chi)$ to be the set of all $x\in G$ such that $|\chi(x)|=\chi(e)$ when $e$ is the identity of the group. For every irreducible charcter we know that

$\frac{|G|}{\chi(e)^2} \leq \frac{1}{\chi(e)} \sum_{x\in G} |\chi(x)| \leq \frac{|G|}{\chi(e)}$.

Now suppose that ${\bf Z}(\chi)$ is trivial i.e. it includes only $e$. Can we find a better bound for $\frac{1}{\chi(e)} \sum_{x\in G} |\chi(x)|$?

My firs clue about this question was in observance of affine $p$-groups: they satisfy this condition: (${\bf Z}(\chi)$ is trivial for only non-linear character $\chi_\pi$ of affine $p$-groups). On the other hand, for this non-linear character,

$\frac{1}{\chi_\pi(e)} \sum_{x\in G} |\chi_\pi(x)| =2$

for every $p$.

Subsequently, I tried some more character tables for finite groups that some of their characters, say $\chi$, satisify this condition and I observed that $\frac{1}{\chi(e)} \sum_{x\in G} |\chi(x)|$ is so close to $\frac{|G|}{\chi(e)^2}$ than $\frac{|G|}{\chi(e)}$.