The Expectation Value of $L_z$

What is the expectation value of $L_z$ in the state $\psi(\vec{r})=R(r)(\sqrt{2\over 3}Y_{11}(\theta,\phi)-i\sqrt{1\over 3}Y_{1-1}(\theta,\phi))$?

The radial part plays no role. The angular part is properly normalized.

$$\begin{eqnarray*} \langle\psi\vert L_z\vert\psi\rangle&=&\left<\sqrt{2\over 3}Y... ... &=&\left({2\over 3}-{1\over 3}\right)\hbar={1\over 3}\hbar \\ \end{eqnarray*}$$