Dirichlet series
| L(χ,s) = 1 | + (0.898 − 0.439i)2-s + (0.983 + 0.181i)3-s + (0.613 − 0.789i)4-s + (0.113 − 0.993i)5-s + (0.962 − 0.269i)6-s + (−0.715 + 0.699i)7-s + (0.203 − 0.979i)8-s + (0.934 + 0.356i)9-s + (−0.334 − 0.942i)10-s + (−0.998 + 0.0455i)11-s + (0.746 − 0.665i)12-s + (−0.648 + 0.761i)13-s + (−0.334 + 0.942i)14-s + (0.291 − 0.956i)15-s + (−0.247 − 0.968i)16-s + (0.0227 + 0.999i)17-s + ⋯ |
| L(s,χ) = 1 | + (0.898 − 0.439i)2-s + (0.983 + 0.181i)3-s + (0.613 − 0.789i)4-s + (0.113 − 0.993i)5-s + (0.962 − 0.269i)6-s + (−0.715 + 0.699i)7-s + (0.203 − 0.979i)8-s + (0.934 + 0.356i)9-s + (−0.334 − 0.942i)10-s + (−0.998 + 0.0455i)11-s + (0.746 − 0.665i)12-s + (−0.648 + 0.761i)13-s + (−0.334 + 0.942i)14-s + (0.291 − 0.956i)15-s + (−0.247 − 0.968i)16-s + (0.0227 + 0.999i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr
=\mathstrut & (0.661 - 0.750i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr
=\mathstrut & (0.661 - 0.750i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(139\) |
| \( \varepsilon \) | = | $0.661 - 0.750i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{139} (30, \cdot )$ |
| Sato-Tate | : | $\mu(69)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 139,\ (0:\ ),\ 0.661 - 0.750i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $1.993701086 - 0.9000447968i$ |
| $L(\frac12,\chi)$ | $\approx$ | $1.993701086 - 0.9000447968i$ |
| $L(\chi,1)$ | $\approx$ | 1.895548210 - 0.5961960944i |
| $L(1,\chi)$ | $\approx$ | 1.895548210 - 0.5961960944i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]