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Abstract

In this thesis we present a new method to construct modular forms using rational functions. It relies on contour integration and Weil's converse theorem. We give several applications, reaching from a relation between cotangent sums and $L$-functions, formulas for Eichler integrals and period polynomials and series representations for $L$-functions corresponding to products of Eisenstein series. \\ With similar ideas, based on contour integration, we move on to equations which were originally studied by Ramanujan and generalize his formulas to those containing $L$-functions at rational arguments. We work out a very general framework for finding new equations of the Ramanujan type that can be applied to a wide range of $L$-functions.