  We study a modular function $\Lambda_{k,\ell}$ which is one of generalized
$\lambda$ functions. We show $\Lambda_{k,\ell}$ and the modular invariant
function $j$ generate the modular function field with respect to the modular
subgroup $\Gamma_1(N)$. Further we prove that $\Lambda_{k,\ell}$ is integral
over $\mathbf Z[j]$. From these results, we obtain that the value of
$\Lambda_{k,\ell}$ at an imaginary quadratic point is an algebraic integer and
generates a ray class field over the Hilbert class field.
