# nt. number theory – Functional equation of the L function of the twisted triple product

Leave $$mathbb {E} = E_1 times E_2 times E_3$$ denote the product of three elliptical curves on $$mathbb {Q}$$ first level $$p$$ and consider the $$p$$-adical representation of Galois $$V_p ( mathbb {E}) = H ^ 1_ {et} (E_ {1 / bar { mathbb {Q}}}, mathbb {Q} _p) otimes H ^ 1_ {et} (E_ {2 / bar { mathbb {Q}}}, mathbb {Q} _p) otimes H ^ 1_ {et} (E_ {3 / bar { mathbb {Q}}}, mathbb {Q} _p).$$ We denote by $$L ( mathbb {E}, s) = L (V_p ( mathbb {E}), s)$$ the associated triple product $$L$$-function. It has a functional equation centered on $$s = 2$$ with global sign equal to $$a_p (E_1) a_p (E_2) a_p (E_2) in { pm 1 }$$ (cf. Gross-Kudla & # 39; 92). Here, $$a_p (E_i)$$ denotes the $$p$$-th Fourier coefficient of weight 2 normalized new level form $$Gamma_0 (p)$$ associated to $$E_i$$ by modularity

Leave $$chi$$ be a Dirichlet character module $$p$$ 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 your global sign?

Thanks in advance for any assistance.