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.