If $Z\sim N(0,1)$, find distribution of $Z^2$ using moment generating functions

Question

Let $Z\sim N(0,1)$ and $Y=Z^2$. Find the distribution of $Y$ using moment generating functions. \begin{align*} M_Y(t)&=E(e^{Z^2t})\\ &=\int_{-\infty}^\infty e^{z^2t}f_Z(z)\ dz\\ &=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{z^2t}\cdot e^{-\frac{z^2}{2}}\ dz\\ &=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{z^2(t-\frac12)}\ dz \end{align*} I'm not sure how to simplify this or find which distribution this is