Subconvexity for a double Dirichlet series

For two real characters ; 0 of conductor dividing 8 de ne Z(s; w; ; 0) := 2(2s + 2w 􀀀 1) X d odd L2(s; d ) 0(d) dw where 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 D12. The convexity bound for Z(s; w; ; 0) is jsw(s + w)j1=4+" for <s = <w = 1=2. It is proved that Z(s; w; ; 0) jsw(s + w)j1=6+"; <s = <w = 1=2: Moreover, the following mean square Lindel of-type bound holds: Z Y1 􀀀Y1 Z Y2 􀀀Y2 jZ(1=2 + it; 1=2 + iu; ; 0)j2 du dt (Y1Y2)1+"; for any Y1; Y2 > 1.