\langle l_z \rangle =\int d^3\vec{r} \psi^*(\vec{r})\hat{l_z}\psi(\vec{r})=\sum_{mn}c_m c_n\int d\phi\psi_m^*(\phi)\hat{l}_z\phi_n(\phi)=c\sum_{mn}(l_z)_n\int d\phi \psi_m^*(\phi)\psi_n(\phi)=c\sum_{mn}(l_z)_n\delta_{mn}=c\sum_m(l_z)_m=c(-2\hbar+2\hbar)=0