Adolf Hurwitz’s estate contains a note from the early 1880s on the converse to Riemann’s proof of the functional equation for the zeta-function; this idea has later been elaborated by Hans Hamburger for a characterization of the zeta-function by its functional equation and by Eugène Cahen and Erich Hecke with respect to modular forms. In this note, we present Hurwitz’s reasoning and comment on the historical context.