  We categorify the Hecke L-functions of $\mathrm{GL}(1)$ by replacing the
L-functions with "modules of zeta integrals". These modules of zeta integrals
are generated by the classical L-function. This approach allows us to
categorify questions regarding L-functions, as well as make their construction
more canonical by avoiding the GCD procedure usually used to define them.
  Let $F$ be a number field, and let $\mathfrak{X}(F)$ be the space of Hecke
characters on $F$. We define a ring $\mathscr{S}$ of holomorphic functions on
$\mathfrak{X}(F)$ and an $\mathscr{S}$-module $\mathscr{L}$ of zeta integrals
on $\mathfrak{X}(F)$. Using a canonical trivialization of the line bundle
$\mathscr{L}$ in some localization $\mathscr{S}'$ of $\mathscr{S}$, we show
that the module of zeta integrals $\mathscr{L}$ contains the same information
as the Hecke L-function of $\mathrm{GL}(1)$. Here, the L-function is thought of
as a single function on $\mathfrak{X}(F)$.
  In the paper arXiv:2011.03313, the author has already implicitly used this
idea to derive non-trivial applications for the theory of automorphic
representations. The goal of this paper make the notion of "module of zeta
integrals" explicit and rigorous for future applications.
  The end result is very closely related to a construction by Connes and Meyer,
although seen from an alternative point of view.
  This paper is based on a part of the author's thesis.
