Dirichlet series
| L(χ,s) = 1 | + (0.766 − 0.642i)2-s + (0.686 + 0.727i)3-s + (0.173 − 0.984i)4-s + (0.448 − 0.893i)5-s + (0.993 + 0.116i)6-s + (−0.835 + 0.549i)7-s + (−0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (−0.230 − 0.973i)10-s + (0.230 − 0.973i)11-s + (0.835 − 0.549i)12-s + (−0.835 + 0.549i)13-s + (−0.286 + 0.957i)14-s + (0.957 − 0.286i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯ |
| L(s,χ) = 1 | + (0.766 − 0.642i)2-s + (0.686 + 0.727i)3-s + (0.173 − 0.984i)4-s + (0.448 − 0.893i)5-s + (0.993 + 0.116i)6-s + (−0.835 + 0.549i)7-s + (−0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (−0.230 − 0.973i)10-s + (0.230 − 0.973i)11-s + (0.835 − 0.549i)12-s + (−0.835 + 0.549i)13-s + (−0.286 + 0.957i)14-s + (0.957 − 0.286i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr
=\mathstrut & (-0.983 + 0.183i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr
=\mathstrut & (-0.983 + 0.183i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(4033\) = \(37 \cdot 109\) |
| \( \varepsilon \) | = | $-0.983 + 0.183i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{4033} (1842, \cdot )$ |
| Sato-Tate | : | $\mu(108)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 4033,\ (0:\ ),\ -0.983 + 0.183i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $-0.1050153949 - 1.136454258i$ |
| $L(\frac12,\chi)$ | $\approx$ | $-0.1050153949 - 1.136454258i$ |
| $L(\chi,1)$ | $\approx$ | 1.363234765 - 0.5982206425i |
| $L(1,\chi)$ | $\approx$ | 1.363234765 - 0.5982206425i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]