The following integral comes out of an expression $\langle |Y_{l,m}(\theta, \phi)|^2\rangle$ over a orientation probability distribution: $$\int_{0}^{2\pi} \int_{0}^{\pi} Y_{lm}^2(\theta, \phi)Y_{l'm'}(\theta, \phi) \sin(\theta) d\theta d\phi$$
When $l = 1,m=1$ and $l' = 2,m' = 0$, the integral is $-\frac{1}{2\sqrt{5\pi}}$. For many other combinations, it is just $0$. What would the general formula for this integral be?