Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2B / Section B / 35 / b / i / Solution

Normalization of a wavefunction on the unit sphere uses solid angle $d\Omega =\sin \theta d\theta d\varphi$, not the flat measure $d\theta d\varphi$. Since $|e^{-i\varphi }|=1$,where $u=\cos \theta$. ThereforeWith the usual phase convention this is the spherical harmonic $Y_{1,-1}$. An arbitrary global unit-modulus factor would not change normalization or any observable.