Roots of unity

Let $\omega$ be a complex cube root of unity with $\omega\ne1$. A fair die is thrown three times. If $r_1$, $r_2$ and $r_3$ are the numbers obtained on the die, then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is

Let $\omega$ be the complex number $\cos\dfrac{2\pi}3+i\sin\dfrac{2\pi}3$. Then the number of distinct complex numbers z satisfying $\begin{vmatrix}z+1&\omega&\omega^2\\\omega&z+\omega^2&1\\\omega^2&1&z+\omega\end{vmatrix}=0$ is equal to