Tate's conjectural analytic continuation and functional equation for Artin-twisted elliptic-curve L-functions

Let $E$) be an elliptic curve and let $\rho$ be an Artin representation, both defined over $\mathbb{Q}$. Define the twisted $L$-function by

$$L(E,\rho,s)=\prod_{q\ \mathrm{prime}}P_q(E,\rho,q^{-s})^{-1},$$

where

$$P_q(E,\rho,T)=\det\left(1-\mathrm{Frob}_q^{-1}\cdot T\mid H_l^1(E)\otimes_{\overline{\mathbb{Q}}_l}V_{\rho,l}\right),$$

$H_l^1(E)$ is the dual of the $l$-adic Tate module of $E$, and $V_{\rho,l}$ is obtained from $\rho$ by extending scalars to $\overline{\mathbb{Q}}_l$. Set

$$L_\infty(s)=\left(2(2\pi)^{-s}\Gamma(s)\right)^{\dim\rho},\qquad \widehat{L}(E,\rho,s)=L_\infty(s)L(E,\rho,s),$$

and let $N(E,\rho)$ denote the conductor of the twist of $E$ by $\rho$. Tate's conjecture. The completed $L$-function $\widehat{L}(E,\rho,s)$ has an analytic continuation to the whole complex plane and satisfies

$$\widehat{L}(E,\rho,s)=w(E,\rho)N(E,\rho)^{1-s}\widehat{L}(E,\rho^*,2-s),$$

where $\rho^*$ is the contragredient representation of $\rho$, and the root number $w(E,\rho)$ is a complex number of absolute value $1$.