We defined Eisenstein series as a special case of Poincaré series in §2.6 for weight k ≧ 3. On the other hand, we also constructed the space of Eisenstein series by modular forms corresponding to products of two Dirichlet L-functions in §4.7. In this chapter, we further investigate Eisenstein series. Though the general arguments in §7.2 are applicable to any weight k, we explain in §7.1 the case of weight k ≧ 3 separately, since that case is easy to handle because of the convergence of the series. In §7.2, we generalize the notion of Eisenstein series and define Eisenstein series with a complex parameter s. We calculate the Fourier expansions of these Eisenstein series and obtain the analytic continuation on parameter s following [Shimura 9, 12].
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