Let $\mathbb{E}=E_1\times E_2\times E_3$ denote the product of three elliptic curves over $\mathbb{Q}$ of prime level $p$ and consider the $p$-adic Galois representation $$V_p(\mathbb{E})=H^1_{et}(E_{1/\bar{\mathbb{Q}}}, \mathbb{Q}_p)\otimes H^1_{et}(E_{2/\bar{\mathbb{Q}}}, \mathbb{Q}_p)\otimes H^1_{et}(E_{3/\bar{\mathbb{Q}}}, \mathbb{Q}_p).$$ Denote by $L(\mathbb{E}, s)=L(V_p(\mathbb{E}),s)$ the associated triple product $L$-function. It has a functional equation centered at $s=2$ with global sign equal to $a_p(E_1)a_p(E_2)a_p(E_2)\in \{ \pm 1 \}$ (cf. Gross-Kudla '92). Here, $a_p(E_i)$ denotes the $p$-th Fourier coefficient of the weight 2 normalized newform of level $\Gamma_0(p)$ associated to $E_i$ by modularity.

Let $\chi$ be a Dirichlet character modulo $p$ and denote by $L(\mathbb{E}\otimes \chi, s)$ the $L$-function attached to the Galois representation $V_p(\mathbb{E})\otimes \chi$. My question is: what is the functional equation of $L(\mathbb{E}\otimes \chi, s)$ and what is its global sign?

Thank you in advance for any help.