Dirichlet series
| L(χ,s) = 1 | + (0.984 − 0.173i)2-s + (0.893 + 0.448i)3-s + (0.939 − 0.342i)4-s + (−0.396 − 0.918i)5-s + (0.957 + 0.286i)6-s + (−0.993 − 0.116i)7-s + (0.866 − 0.5i)8-s + (0.597 + 0.802i)9-s + (−0.549 − 0.835i)10-s + (−0.549 + 0.835i)11-s + (0.993 + 0.116i)12-s + (0.116 − 0.993i)13-s + (−0.998 + 0.0581i)14-s + (0.0581 − 0.998i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + ⋯ |
| L(s,χ) = 1 | + (0.984 − 0.173i)2-s + (0.893 + 0.448i)3-s + (0.939 − 0.342i)4-s + (−0.396 − 0.918i)5-s + (0.957 + 0.286i)6-s + (−0.993 − 0.116i)7-s + (0.866 − 0.5i)8-s + (0.597 + 0.802i)9-s + (−0.549 − 0.835i)10-s + (−0.549 + 0.835i)11-s + (0.993 + 0.116i)12-s + (0.116 − 0.993i)13-s + (−0.998 + 0.0581i)14-s + (0.0581 − 0.998i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr
=\mathstrut & (0.617 + 0.786i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr
=\mathstrut & (0.617 + 0.786i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(4033\) = \(37 \cdot 109\) |
| \( \varepsilon \) | = | $0.617 + 0.786i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{4033} (1380, \cdot )$ |
| Sato-Tate | : | $\mu(108)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 4033,\ (1:\ ),\ 0.617 + 0.786i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $3.828771614 + 1.862817372i$ |
| $L(\frac12,\chi)$ | $\approx$ | $3.828771614 + 1.862817372i$ |
| $L(\chi,1)$ | $\approx$ | 2.096481366 - 0.08658057169i |
| $L(1,\chi)$ | $\approx$ | 2.096481366 - 0.08658057169i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]