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
\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]