Dirichlet series
| L(χ,s) = 1 | + i·3-s − i·5-s − i·7-s − 9-s − i·11-s + 13-s + 15-s + 19-s + 21-s − i·23-s − 25-s − i·27-s − i·29-s + i·31-s + 33-s + ⋯ |
| L(s,χ) = 1 | + i·3-s − i·5-s − i·7-s − 9-s − i·11-s + 13-s + 15-s + 19-s + 21-s − i·23-s − 25-s − i·27-s − i·29-s + i·31-s + 33-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr
=\mathstrut & (0.615 - 0.788i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr
=\mathstrut & (0.615 - 0.788i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(68\) = \(2^{2} \cdot 17\) |
| \( \varepsilon \) | = | $0.615 - 0.788i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{68} (55, \cdot )$ |
| Sato-Tate | : | $\mu(4)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 68,\ (1:\ ),\ 0.615 - 0.788i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $1.293981416 - 0.6313702366i$ |
| $L(\frac12,\chi)$ | $\approx$ | $1.293981416 - 0.6313702366i$ |
| $L(\chi,1)$ | $\approx$ | 1.069483852 - 0.1316594787i |
| $L(1,\chi)$ | $\approx$ | 1.069483852 - 0.1316594787i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]