Hello, I am very lost with the following problem, is there a neat way of doing it, or just solving integral after integral is the way to go?

Any hint would be greatly appreciated.

Let $\displaystyle X \sim N_p(0,\Sigma)$ be independent of $\displaystyle Z \sim N(0,1)$. Show that $\displaystyle Y=X/Z$ has characteristic function

$\displaystyle \varphi_Y(t) = \exp\{-(t^T \Sigma t)^{1/2}\}$

Thanks in advance,