Dirichlet series
| L(χ,s) = 1 | + (−0.248 − 0.968i)2-s + (0.998 − 0.0627i)3-s + (−0.876 + 0.481i)4-s + (0.968 + 0.248i)5-s + (−0.309 − 0.951i)6-s + (−0.770 + 0.637i)7-s + (0.684 + 0.728i)8-s + (0.992 − 0.125i)9-s − i·10-s + (0.125 + 0.992i)11-s + (−0.844 + 0.535i)12-s + (0.637 − 0.770i)13-s + (0.809 + 0.587i)14-s + (0.982 + 0.187i)15-s + (0.535 − 0.844i)16-s + (−0.309 + 0.951i)17-s + ⋯ |
| L(s,χ) = 1 | + (−0.248 − 0.968i)2-s + (0.998 − 0.0627i)3-s + (−0.876 + 0.481i)4-s + (0.968 + 0.248i)5-s + (−0.309 − 0.951i)6-s + (−0.770 + 0.637i)7-s + (0.684 + 0.728i)8-s + (0.992 − 0.125i)9-s − i·10-s + (0.125 + 0.992i)11-s + (−0.844 + 0.535i)12-s + (0.637 − 0.770i)13-s + (0.809 + 0.587i)14-s + (0.982 + 0.187i)15-s + (0.535 − 0.844i)16-s + (−0.309 + 0.951i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr
=\mathstrut & (0.941 - 0.338i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr
=\mathstrut & (0.941 - 0.338i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(101\) |
| \( \varepsilon \) | = | $0.941 - 0.338i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{101} (12, \cdot )$ |
| Sato-Tate | : | $\mu(100)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 101,\ (1:\ ),\ 0.941 - 0.338i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $2.186603091 - 0.3808328061i$ |
| $L(\frac12,\chi)$ | $\approx$ | $2.186603091 - 0.3808328061i$ |
| $L(\chi,1)$ | $\approx$ | 1.385218825 - 0.3347323994i |
| $L(1,\chi)$ | $\approx$ | 1.385218825 - 0.3347323994i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]