Dirichlet series
| L(χ,s) = 1 | + (−0.587 − 0.809i)2-s + (0.809 − 0.587i)3-s + (−0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.951 − 0.309i)6-s + (0.587 − 0.809i)7-s + (0.951 − 0.309i)8-s + (0.309 − 0.951i)9-s + (0.951 − 0.309i)10-s − i·11-s + (0.309 + 0.951i)12-s + 13-s − 14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (−0.951 − 0.309i)17-s + ⋯ |
| L(s,χ) = 1 | + (−0.587 − 0.809i)2-s + (0.809 − 0.587i)3-s + (−0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.951 − 0.309i)6-s + (0.587 − 0.809i)7-s + (0.951 − 0.309i)8-s + (0.309 − 0.951i)9-s + (0.951 − 0.309i)10-s − i·11-s + (0.309 + 0.951i)12-s + 13-s − 14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (−0.951 − 0.309i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr
=\mathstrut & (-0.180 - 0.983i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr
=\mathstrut & (-0.180 - 0.983i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(61\) |
| \( \varepsilon \) | = | $-0.180 - 0.983i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{61} (37, \cdot )$ |
| Sato-Tate | : | $\mu(20)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 61,\ (1:\ ),\ -0.180 - 0.983i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $0.9605985573 - 1.153378356i$ |
| $L(\frac12,\chi)$ | $\approx$ | $0.9605985573 - 1.153378356i$ |
| $L(\chi,1)$ | $\approx$ | 0.9200223296 - 0.5781922535i |
| $L(1,\chi)$ | $\approx$ | 0.9200223296 - 0.5781922535i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]