  For every odd prime $p$, we exhibit families of irreducible Artin
representations $\tau$ with the property that for every elliptic curve $E$ the
order of the zero of the twisted $L$-function $L(E,\tau,s)$ at $s\!=\!1$ must
be a multiple~of~$p$. Analogously, the multiplicity of $\tau$ in the Selmer
group of $E$ must also be divisible by $p$. We give further examples where
$\tau$ can moreover be twisted by any character that factors through the
$p$-cyclotomic extension, and examples where the $L$-functions are those of
twists of certain Hilbert modular forms by Dirichlet charaters. These results
are conjectural, and rely on a standard generalisation of the
Birch--Swinnerton-Dyer conjecture. Our main tool is the theory of Schur indices
from representation theory.
