Dirichlet series
| L(χ,s) = 1 | + (0.457 + 0.889i)2-s + (−0.252 − 0.967i)3-s + (−0.581 + 0.813i)4-s + (0.322 − 0.946i)5-s + (0.744 − 0.667i)6-s + (−0.905 − 0.424i)7-s + (−0.989 − 0.145i)8-s + (−0.872 + 0.489i)9-s + (0.989 − 0.145i)10-s + (0.997 + 0.0729i)11-s + (0.934 + 0.357i)12-s + (−0.457 − 0.889i)13-s + (−0.0365 − 0.999i)14-s + (−0.997 − 0.0729i)15-s + (−0.322 − 0.946i)16-s + (−0.109 − 0.994i)17-s + ⋯ |
| L(s,χ) = 1 | + (0.457 + 0.889i)2-s + (−0.252 − 0.967i)3-s + (−0.581 + 0.813i)4-s + (0.322 − 0.946i)5-s + (0.744 − 0.667i)6-s + (−0.905 − 0.424i)7-s + (−0.989 − 0.145i)8-s + (−0.872 + 0.489i)9-s + (0.989 − 0.145i)10-s + (0.997 + 0.0729i)11-s + (0.934 + 0.357i)12-s + (−0.457 − 0.889i)13-s + (−0.0365 − 0.999i)14-s + (−0.997 − 0.0729i)15-s + (−0.322 − 0.946i)16-s + (−0.109 − 0.994i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr
=\mathstrut & (0.368 - 0.929i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr
=\mathstrut & (0.368 - 0.929i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(173\) |
| \( \varepsilon \) | = | $0.368 - 0.929i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{173} (24, \cdot )$ |
| Sato-Tate | : | $\mu(86)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 173,\ (0:\ ),\ 0.368 - 0.929i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $0.8033933081 - 0.5457024607i$ |
| $L(\frac12,\chi)$ | $\approx$ | $0.8033933081 - 0.5457024607i$ |
| $L(\chi,1)$ | $\approx$ | 0.9911835121 - 0.1491675089i |
| $L(1,\chi)$ | $\approx$ | 0.9911835121 - 0.1491675089i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]