A Sato-Tate group $G$ of degree $d$ is said to be **rational** if for every irreducible character $\chi$ of $\GL_d(\C)$ and every component $H$ of $G$ we have
$$
\int_H \chi(g)\mu_H \in \Z,
$$
where $\mu_H$ denotes the restriction to $H$ of the Haar measure $\mu$ on $G$, normalized so that $\mu_H(H)=1$. If $G$ is the Sato-Tate group of an arithmetic L-function
$$
L(s)=\prod_{\mathfrak p}L_\mathfrak{p}(N(\mathfrak{p})^{-s})^{-1},
$$
this means that the polynomials $L_{\mathfrak p}(T)$ have integer coefficients, equivalently, the $L$-function $L(s)$ is rational.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2021-01-01 15:02:58

**Referred to by:**

- dq.st.source
- rcs.cande.st_group
- rcs.rigor.st_group
- rcs.source.st_group
- st_group.subsupgroups
- st_group.summary
- st_group.supgroups
- lmfdb/sato_tate_groups/main.py (line 406)
- lmfdb/sato_tate_groups/main.py (line 625)
- lmfdb/sato_tate_groups/main.py (line 1080)
- lmfdb/sato_tate_groups/main.py (line 1220)
- lmfdb/sato_tate_groups/templates/st_display.html (line 11)

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

- 2021-01-01 15:02:58 by Andrew Sutherland (Reviewed)
- 2021-01-01 15:02:19 by Andrew Sutherland
- 2018-06-20 04:07:30 by Kiran S. Kedlaya (Reviewed)

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