  We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are
linear forms in the zeta values $\zeta(n)$. They naturally include the
Beukers-Rhin-Viola integrals for $\zeta(2)$ and the Ball-Rivoal linear forms in
odd zeta values. We give a general integral formula for the coefficients of the
linear forms and a geometric interpretation of the vanishing of the
coefficients of a given parity. The main underlying result is a geometric
construction of a minimal ind-object in the category of mixed Tate motives over
$\mathbb{Z}$ which contains all the non-trivial extensions between simple
objects. In a joint appendix with Don Zagier, we prove the compatibility
between the structure of the motives considered here and the representations of
their periods as sums of series.
