Is there a Schwartz function $\zeta(t)$, defined on $\mathbb{R}$, satisfying the following:

1. $\int \zeta(t)\: dt=1$,
2. $\int t^k \zeta(t)\: dt=0$ for all $k\geq 1$,
3. $\mathrm{supp}(\zeta)\subset (0,\infty)$.

If we drop the third property, this is easy! Just let $\zeta$ be the Fourier transform of a Schwartz function which equals $1$ on a neighborhood of $0$.