The tensor product of two states with spin can be decomposed into irreducible representations via Clebsh-Gordan coefficients
$$|j_1, m_1, j_2, m_2 \rangle = \sum C^{JM}_{j_1, m_1, j_2, m_2} |JM\rangle\,.$$
Since spherical harmonics $Y_{\ell m}$ are representations of $SO(3)$, I would have expected a similar decomposition, i.e.
$$Y_{\ell_1 m_1} (\Omega) Y_{\ell_2 m_2}(\Omega) = \sum C^{L M}_{\ell_1 m_1 \ell_2 m_2} Y_{L M}(\Omega)\,.$$
However, the Wikipedia page on Clebsh-Gordan coefficients instead gives the expansion
$$Y_{\ell_1 m_1} (\Omega) Y_{\ell_2 m_2}(\Omega) = \sum_{L,M} \sqrt{\frac{(2\ell_1 + 1)(2\ell_2 + 1)}{4\pi (2 L+1)}} C^{L M}_{\ell_1 m_1 \ell_2 m_2}C^{L 0}_{\ell_1 0 \ell_2 0} Y_{L M}(\Omega)\,.$$
How can I understand where these additional terms come from? I've found some derivations of the expression in Sakurai's Modern Quantum Mechanics, and I can follow the derivation, but I don't understand where the discrepancy arises on the level of representation theory.