  Under suitable, fairly weak hypotheses on an elliptic curve $E/\mathbb{Q}$
and a primitive non-trivial Dirichlet character $\chi$, we show that the
algebraic $L$-value $\mathscr{L}(E,\chi)$ at $s=1$ is an algebraic integer. For
instance, for semistable curves $\mathscr{L}(E,\chi)$ is integral whenever $E$
admits no isogenies defined over $\mathbb{Q}$. Moreover we give examples
illustrating that our hypotheses are necessary for integrality to hold.
