We prove a general result relating the shape of the Euler product of an$L$-function to the analytic properties of certain linear twists of the$L$-function itself. Then, by a sharp form of the transformation formula forlinear twists, we check the required analytic properties in the case of$L$-functions of degree 2 and conductor 1 in the Selberg class. Finally weprove a converse theorem, showing that $zeta(s)^2$ is the only member of theSelberg class satisfying the above conditions and, moreover, having a pole at$s=1$.
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机译：我们证明了一般结果与$ L $ -Function本身的某些线性扭曲的分析属性相关联的欧拉产品的形状。然后，通过尖锐的转换公式的尖锐形式，我们在Selberg类中的$ 2和指挥1的$ L $ -Functions的情况下检查所需的分析性质。最后，Weprove悔改定理，表明$ Zeta（s）^ 2 $是满足上述条件的唯一一体的Theselberg类成员，而且，杆子以$ s = 1 $。
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