Subconvexity for a double Dirichlet series

For two real characters ψ , ψ′ of conductor dividing 8 define where χ d = (d/.) and the subscript 2 denotes the fact that the Euler factor at 2 has been removed. These double Dirichlet series can be extended to C² possessing a group of functional equations isomorphic to D 12 . The convexity bound for Z ( s , w ; ψ , ψ′ ) is | sw ( s + w )| 1/4+ε for ℜ s =ℜ w =1/2. It is proved that Moreover, the following mean square Lindelöf-type bound holds: for any Y 1 , Y 2 ≥1.