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 $Q(\sqrt[p^n]{m},\mu_{p^n})/Q$, 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 $\Omega_{+}^{d^+}|\Omega_{-}^{d^-}|\epsilon(\tau)$ is algebraic and Galois-equivariant, as predicted by Deligne's conjecture.