Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties
- **Symmetry:** $\zeta \overset{d}{=} - \zeta$.
- **Small-ball probability:**  there exists a constant $\alpha > 0$ such that $P(\|\zeta^\otimes\| \le u\sqrt{n}) \le (\alpha u)^n$, for all $u \ge 0$ and positive integer $n$. Note that $\zeta^\otimes$  is a random vector on $\mathbb R^n$ with iid coordinates having the same distribution as $\zeta$.

Of course, $\zeta \sim N(0,1)$ fits the bill. I'm looking for general characterization.