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Apr 15, 2013 at 1:50 comment added Mark Lewko Of course showing anything implies (1) seems out of reach at present. But there is some very interesting partial progress relating (2) and (3). In particular, there is a line of work started with Conrey and Farmer which shows (roughly) that if a Dirichlet series has a functional equation and Euler product (of a certain form) then it must arise from a modular form. See: ams.org/mathscinet-getitem?mr=1360623
Apr 14, 2013 at 23:13 comment added user9072 I hoped you would also say something on this question. In fact, I meant to mention your work, but then got scared by the idea of misrepresenting it 'in your presence.'
Apr 14, 2013 at 17:28 history answered Frank Thorne CC BY-SA 3.0