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

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

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