Dirichlet series
| L(χ,s) = 1 | + (−0.707 − 0.707i)2-s + (−0.382 + 0.923i)3-s + i·4-s + (0.923 − 0.382i)6-s + (−0.923 + 0.382i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.923 − 0.382i)12-s − 13-s + (0.923 + 0.382i)14-s − 16-s + i·18-s + (−0.707 + 0.707i)19-s − i·21-s + (0.923 + 0.382i)22-s + ⋯ |
| L(s,χ) = 1 | + (−0.707 − 0.707i)2-s + (−0.382 + 0.923i)3-s + i·4-s + (0.923 − 0.382i)6-s + (−0.923 + 0.382i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.923 − 0.382i)12-s − 13-s + (0.923 + 0.382i)14-s − 16-s + i·18-s + (−0.707 + 0.707i)19-s − i·21-s + (0.923 + 0.382i)22-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr
=\mathstrut & (-0.724 + 0.688i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr
=\mathstrut & (-0.724 + 0.688i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(85\) = \(5 \cdot 17\) |
| \( \varepsilon \) | = | $-0.724 + 0.688i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{85} (3, \cdot )$ |
| Sato-Tate | : | $\mu(16)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 85,\ (0:\ ),\ -0.724 + 0.688i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $0.08928106553 + 0.2235191158i$ |
| $L(\frac12,\chi)$ | $\approx$ | $0.08928106553 + 0.2235191158i$ |
| $L(\chi,1)$ | $\approx$ | 0.4294688579 + 0.1020763067i |
| $L(1,\chi)$ | $\approx$ | 0.4294688579 + 0.1020763067i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]