Dirichlet series
| L(χ,s) = 1 | + (−0.841 + 0.540i)3-s + (−0.415 − 0.909i)5-s + (−0.959 + 0.281i)7-s + (0.415 − 0.909i)9-s + (−0.654 − 0.755i)11-s + (−0.959 − 0.281i)13-s + (0.841 + 0.540i)15-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (0.654 − 0.755i)21-s + (−0.654 + 0.755i)25-s + (0.142 + 0.989i)27-s + (−0.142 + 0.989i)29-s + (−0.841 − 0.540i)31-s + (0.959 + 0.281i)33-s + ⋯ |
| L(s,χ) = 1 | + (−0.841 + 0.540i)3-s + (−0.415 − 0.909i)5-s + (−0.959 + 0.281i)7-s + (0.415 − 0.909i)9-s + (−0.654 − 0.755i)11-s + (−0.959 − 0.281i)13-s + (0.841 + 0.540i)15-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (0.654 − 0.755i)21-s + (−0.654 + 0.755i)25-s + (0.142 + 0.989i)27-s + (−0.142 + 0.989i)29-s + (−0.841 − 0.540i)31-s + (0.959 + 0.281i)33-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr
=\mathstrut & (-0.529 - 0.848i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr
=\mathstrut & (-0.529 - 0.848i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(92\) = \(2^{2} \cdot 23\) |
| \( \varepsilon \) | = | $-0.529 - 0.848i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{92} (19, \cdot )$ |
| Sato-Tate | : | $\mu(22)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 92,\ (0:\ ),\ -0.529 - 0.848i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $0.1551232152 - 0.2796552153i$ |
| $L(\frac12,\chi)$ | $\approx$ | $0.1551232152 - 0.2796552153i$ |
| $L(\chi,1)$ | $\approx$ | 0.5229382686 - 0.1117682732i |
| $L(1,\chi)$ | $\approx$ | 0.5229382686 - 0.1117682732i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]