Motivic weight (reviewed)

The motivic weight (or arithmetic weight) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$.

If the L-function arises from a motive, then the weight of the motive has the same parity as the motivic weight of the L-function, but the weight of the motive could be larger. This apparent discrepancy comes from the fact that a Tate twist increases the weight of the motive. This corresponds to the change of variables $s \mapsto s + j$ in the L-function of the motive.