Dirichlet series
| L(χ,s) = 1 | − 2-s + (0.286 − 0.957i)3-s + 4-s + (−0.448 + 0.893i)5-s + (−0.286 + 0.957i)6-s + (−0.835 + 0.549i)7-s − 8-s + (−0.835 − 0.549i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (0.286 − 0.957i)12-s + (−0.893 − 0.448i)13-s + (0.835 − 0.549i)14-s + (0.727 + 0.686i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
| L(s,χ) = 1 | − 2-s + (0.286 − 0.957i)3-s + 4-s + (−0.448 + 0.893i)5-s + (−0.286 + 0.957i)6-s + (−0.835 + 0.549i)7-s − 8-s + (−0.835 − 0.549i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (0.286 − 0.957i)12-s + (−0.893 − 0.448i)13-s + (0.835 − 0.549i)14-s + (0.727 + 0.686i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr
=\mathstrut & (0.968 + 0.247i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr
=\mathstrut & (0.968 + 0.247i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(4033\) = \(37 \cdot 109\) |
| \( \varepsilon \) | = | $0.968 + 0.247i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{4033} (2871, \cdot )$ |
| Sato-Tate | : | $\mu(108)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 4033,\ (0:\ ),\ 0.968 + 0.247i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $0.1253052305 + 0.01577920563i$ |
| $L(\frac12,\chi)$ | $\approx$ | $0.1253052305 + 0.01577920563i$ |
| $L(\chi,1)$ | $\approx$ | 0.3964078715 - 0.09829314990i |
| $L(1,\chi)$ | $\approx$ | 0.3964078715 - 0.09829314990i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]