Normal bases of ray class fields over imaginary quadratic fields

We develop a criterion for a normal basis, and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. This result would be an answer for the Lang-Schertz conjecture on a ray class field with modulus generated by an integer ($\geq2$).