We study the solutions of Hecke's functional equation in the framework of the extended Selberg class. It turns out that, notwithstanding certain differences between the two theories, there are strong analogies in the behavior of the dimensions of the spaces of solutions. We also provide details to an old argument of Hecke, which is presented in a sketchy way both in Hecke's original works and in subsequent books on Hecke's theory.
"A note on Hecke's functional equation and the Selberg class." Funct. Approx. Comment. Math. 41 (2) 211 - 220, December 2009. https://doi.org/10.7169/facm/1261157810