Vanishing integral on $(-c,c)$

Question

Assume we have a function $f\colon (-c,c)\to\mathbb{R}$ such that $\int_{-c}^cf(s)\,\mathrm{d} s=0$. Is there a general approach to finding conditions on $f$ that would ensure that $\int_{-c}^c s\cdot f(s)\,\mathrm{d} s$ also vanishes? I wanted to work with odd or even functions since I felt like decomposing $f$ into its odd and even part would be fruitful. Then, obviously, $f$ being even would be enough, since an integral over an odd function about an interval centered at $0$ vanishes. What happens though if $f$ is odd? Seeing that $f(s)=s^{2n+1}$ cannot work since then we have an integral over a nonnegative function that doesn't vanish everywhere, thus the integral cannot vanish, makes me wonder if there is such a general condition on an odd $f$, but for orthogonal polynomials and their three-term recurrence relation it obviously works. Any ideas?