For what values of $k$, $t_{0}$, or $x$ is this integral convergent?

Question

$$I(x)=\int_{-\infty}^{0}{({-\xi})^{1.5k}\sin{(\xi{t_{0})}}e^{-i\xi{x}}}d\xi,$$ $k$ is an integer greater or equal to $0$, $t_{0}\in[0,\infty)$, $x\in(-\infty,\infty)$. I need to find an approximate evaluation of the integral using quadratures, so I need to be sure of convergence.