If $u : \mathbb{R}^3 \to \mathbb{R}$ is a smooth function satisfying $\sum_{m=-l}^l \langle u, Y_{l,m} \rangle_{S^2}=0$, is it necessarily radial?

Question

Let $Y_{l,m} : \mathbb{S}^2 \to \mathbb{R}$ be the spherical harmonics, as presented in wiki.

$u : \mathbb{R}^3 \to \mathbb{R}$ be a smooth function such that \begin{equation} \sum_{m=-l}^l \langle u, Y_{l,m} \rangle_{S^2}=0 \end{equation} for each $l=0,1,2, \cdots$.

Then, is this $u$ necessarily radial? Or what kind of symmetry does such $u$ have?

Could anyone please explain for me?