Algebraicity of L-values for elliptic curves in a false Tate curve tower

Let $E$ be an elliptic curve over $Q$, and $\tau$ an Artin representation over $Q$ that factors through the non-abelian extension $Q(\sqrt[p^n]{m},{\mu }_{p^n})\mathrm{/}Q$, where $p$ is an odd prime and $n,m$ are positive integers. We show that $L(E,\tau ,1)$, the special value at $s=1$ of the $L$-function of the twist of $E$ by $\tau$, divided by the classical transcendental period ${\mathrm{\Omega }}_{+}^{d^{+}}\mathrm{\mid }{\mathrm{\Omega }}_{-}^{d^{-}}\mathrm{\mid }ϵ(\tau )$ is algebraic and Galois-equivariant, as predicted by Deligne’s conjecture.