In this work, we consider the question of whether the critical L values of the quadratic twists of a self-contragredient cuspidal automorphic representation pi over GL2 over a number field determine the representation. It is known that there exist cuspidal automorphic representations pi0 such that L (1/2, pi0 ⊗ chi) = 0 whenever the character chi is quadratic or trivial. So in this case the quadratic twists don't suffice to determine pi0. But we prove that except for the above bad situation, certain quadratic twisted critical L values do determine pi provided the central character of pi is given.; This generalizes the work of W. Luo and D. Ramakrishnan on holomorphic modular forms. In contrast to their approximate functional equation method, we use double Dirichlet series. We rely heavily on the work of B. Fisher and S. Friedberg. The quadratic twists we need are based on the quadratic symbols they define as a generalization of the usual ones. Using these symbols, they construct the double Dirichlet series, which are weighted sums of twisted L-functions of GL2 or GL1 in the variables s or w. These series have the functional equations derived from those of the L-functions, and the functional equations give the series a meromorphic continuation to all of C2 . We refine some of their results for our purposes and sieve the series to get a new series summed over only square free terms, because as usual we cannot handle the complicated correction factors in the definition of the series. Then we will meromorphically continue the new series to the point (s,w) = (1/2,1) by using some estimates and properties of the old series. The analytic data of the new series at the point offers us enough information to get our result.
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