Dirichlet series
| L(χ,s) = 1 | + (−0.382 − 0.923i)3-s + (0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (0.923 − 0.382i)13-s − i·15-s − i·17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + ⋯ |
| L(s,χ) = 1 | + (−0.382 − 0.923i)3-s + (0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (0.923 − 0.382i)13-s − i·15-s − i·17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr
=\mathstrut & (0.773 - 0.634i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr
=\mathstrut & (0.773 - 0.634i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(64\) = \(2^{6}\) |
| \( \varepsilon \) | = | $0.773 - 0.634i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{64} (59, \cdot )$ |
| Sato-Tate | : | $\mu(16)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 64,\ (1:\ ),\ 0.773 - 0.634i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $1.685427568 - 0.6030556268i$ |
| $L(\frac12,\chi)$ | $\approx$ | $1.685427568 - 0.6030556268i$ |
| $L(\chi,1)$ | $\approx$ | 1.221600121 - 0.2742820743i |
| $L(1,\chi)$ | $\approx$ | 1.221600121 - 0.2742820743i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]