Implications of symmetric probability density function

Question

Consider a real-valued random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with probability density function $f$. Suppose $f$ is symmetric around zero. This implies that $\int_{-\infty}^\infty x f(x)dx=0$. Can we say something about $\int_{-\infty}^\infty x (f(x))^2dx$ ?