From the expressions

$$L_{x}=y P_{z}-z P_{y}, \quad L_{y}=z P_{x}-x P_{z}, \quad L_{z}=x P_{y}-y P_{x}$$

show that

$$(x+i y) z$$

is an eigenfunction of $\mathbf{L}^{2}$ and $L_{z}$, and compute the corresponding eigenvalues.