  We consider Laplacian eigenfunctions on a domain $\Omega \subset
\mathbb{R}^d$. Under Neumann boundary conditions, the first eigenfunction is
constant and the others have mean value 0. The situation is different for
Dirichlet boundary conditions: on generic domains, one would expect that every
eigenfunction has nonzero mean value. The other extreme is the ball in
$\mathbb{R}^d$, where among the first $n$ eigenfunctions only $\sim n^{1/d}$
have a mean value different from zero. We prove that in any domain~$\Omega$,
among the first $n$ Dirichlet eigenfunctions at least $ (\log{n})^{-1/2d}
n^{1/2d} $ do \textit{not} have vanishing mean value.
