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.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by David Farmer on 2019-05-07 21:54:05

**Referred to by:**

- columns.lfunc_lfunctions.motivic_weight
- columns.lfunc_search.motivic_weight
- g2c.169.a.169.1.bottom
- lfunction.central_point
- lfunction.coefficient_field
- lfunction.critical_strip
- lfunction.functional_equation
- lfunction.invariants
- rcs.cande.lfunction
- lmfdb/lfunctions/main.py (line 312)
- lmfdb/lfunctions/main.py (line 520)
- lmfdb/lfunctions/templates/Lfunction.html (line 108)
- lmfdb/lfunctions/templates/LfunctionNavigate.html (line 13)

**History:**(expand/hide all)

- 2019-05-07 21:54:05 by David Farmer (Reviewed)
- 2019-05-07 18:45:04 by Andrew Sutherland
- 2019-05-07 11:58:18 by John Voight (Reviewed)
- 2019-05-07 11:57:26 by John Voight
- 2019-05-05 15:34:44 by Andrew Sutherland
- 2019-05-05 15:31:18 by Andrew Sutherland
- 2019-05-05 15:20:56 by Andrew Sutherland (Reviewed)
- 2019-04-30 13:19:55 by David Farmer (Reviewed)
- 2016-05-13 23:15:19 by Andrew Sutherland

**Differences**(show/hide)