  This paper draws connections between the double shuffle equations and
structure of associators; universal mixed elliptic motives as defined by Hain
and Matsumoto; and the Rankin-Selberg method for modular forms for
$SL_2(\mathbb{Z})$. We write down explicit formulae for zeta elements
$\sigma_{2n-1}$ (generators of the Tannaka Lie algebra of the category of mixed
Tate motives over $\mathbb{Z}$) in depths up to four, give applications to the
Broadhurst-Kreimer conjecture, and completely solve the double shuffle
equations for multiple zeta values in depths two and three.
