  The 21-cm signal holds the key to understanding the first structure formation
during cosmic dawn. Theoretical progress over the last decade has focused on
simulations of this signal, given the nonlinear and nonlocal relation between
initial conditions and observables (21-cm or reionization maps). Here, instead,
we propose an {\it effective} and fully analytic model for the 21-cm signal
during cosmic dawn. We take advantage of the exponential-like behavior of the
local star-formation rate density (SFRD) against densities at early times to
analytically find its correlation functions including nonlinearities. The SFRD
acts as the building block to obtain the statistics of radiative fields (X-ray
and Lyman-$\alpha$ fluxes), and therefore the 21-cm signal. We implement this
model as the public Python package Zeus21. This code can fully predict the
21-cm global signal and power spectrum in $\sim 1$ s, with negligible memory
requirements. When comparing against state-of-the-art semi-numerical
simulations from 21CMFAST we find agreement to $\sim10\%$ precision in both the
21-cm global signal and power spectra, after accounting for a (previously
missed) underestimation of adiabatic fluctuations in 21CMFAST. Zeus21 is
modular, allowing the user to vary the astrophysical model for the first
galaxies, and interfaces with the cosmological code CLASS, which enables
searches for beyond standard-model cosmology in 21-cm data. This represents a
step towards bringing 21-cm to the era of precision cosmology.
