Dirichlet series
| L(χ,s) = 1 | + (0.766 + 0.642i)2-s + (−0.835 + 0.549i)3-s + (0.173 + 0.984i)4-s + (0.396 + 0.918i)5-s + (−0.993 − 0.116i)6-s + (−0.993 − 0.116i)7-s + (−0.5 + 0.866i)8-s + (0.396 − 0.918i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (−0.686 − 0.727i)12-s + (0.597 − 0.802i)13-s + (−0.686 − 0.727i)14-s + (−0.835 − 0.549i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯ |
| L(s,χ) = 1 | + (0.766 + 0.642i)2-s + (−0.835 + 0.549i)3-s + (0.173 + 0.984i)4-s + (0.396 + 0.918i)5-s + (−0.993 − 0.116i)6-s + (−0.993 − 0.116i)7-s + (−0.5 + 0.866i)8-s + (0.396 − 0.918i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (−0.686 − 0.727i)12-s + (0.597 − 0.802i)13-s + (−0.686 − 0.727i)14-s + (−0.835 − 0.549i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr
=\mathstrut & (0.472 - 0.881i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr
=\mathstrut & (0.472 - 0.881i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(4033\) = \(37 \cdot 109\) |
| \( \varepsilon \) | = | $0.472 - 0.881i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{4033} (2253, \cdot )$ |
| Sato-Tate | : | $\mu(27)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 4033,\ (0:\ ),\ 0.472 - 0.881i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $-0.2643433080 + 0.1581894142i$ |
| $L(\frac12,\chi)$ | $\approx$ | $-0.2643433080 + 0.1581894142i$ |
| $L(\chi,1)$ | $\approx$ | 0.6638131724 + 0.6824246537i |
| $L(1,\chi)$ | $\approx$ | 0.6638131724 + 0.6824246537i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]