Angular momentum of a radial function times a linear polynomial (${\hat{L}}^2[H(r)(\mathbf{a}\cdot \mathbf{x})]=2{\hslash }^{2}H(r)(\mathbf{a}\cdot \mathbf{x})$)

Orbital angular momentum operators annihilate a radial multiplier $H(r)$. Their squared sum sends every degree-one homogeneous polynomial to $2{\hslash }^2$ times that polynomial, so a nonzero normalizable state $H(r)(\mathbf{a}\cdot \mathbf{x})$ has angular momentum quantum number $l=1$. With real $H$ and real coefficients, its third-component expected value is zero by angular symmetry, though it need not be an eigenstate of that component.