  We study the (mock) Eisenstein series $E_k$ of weight $k \in \{1,3/2,2\}$ for
the Weil representation on an even lattice, defined as the result of Bruinier
and Kuss's coefficient formula for the Eisenstein series naively evaluated at
$k$. We describe the transformation law of $E_k$ in general. Most of this note
is dedicated to collecting examples where the coefficients of $E_k$ contain
interesting arithmetic information. Finally we make a few remarks about the
case $k=1/2$.
