↑

Properties

Label 2-1859-143.43-c0-0-9
Degree $2$
Conductor $1859$
Sign $-0.964 - 0.265i$
Analytic cond. $0.927761$
Root an. cond. $0.963203$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 1.40i)2-s + (0.809 − 1.40i)3-s + (−0.809 − 1.40i)4-s + (−1.30 − 2.26i)6-s + (−0.309 − 0.535i)7-s − 0.999·8-s + (−0.809 − 1.40i)9-s + (−0.5 + 0.866i)11-s − 2.61·12-s − 14-s − 2.61·18-s + (0.809 + 1.40i)19-s − 21-s + (0.809 + 1.40i)22-s + (−0.309 + 0.535i)23-s + (−0.809 + 1.40i)24-s + ⋯
L(s)  = 1  + (0.809 − 1.40i)2-s + (0.809 − 1.40i)3-s + (−0.809 − 1.40i)4-s + (−1.30 − 2.26i)6-s + (−0.309 − 0.535i)7-s − 0.999·8-s + (−0.809 − 1.40i)9-s + (−0.5 + 0.866i)11-s − 2.61·12-s − 14-s − 2.61·18-s + (0.809 + 1.40i)19-s − 21-s + (0.809 + 1.40i)22-s + (−0.309 + 0.535i)23-s + (−0.809 + 1.40i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-0.964 - 0.265i$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1330, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ -0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.092178298\)
\(L(\frac12)\) \(\approx\) \(2.092178298\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.61T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + 1.61T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 0.618T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.204850283684382542838264408034, −8.039713348990108212930832621606, −7.52220539612015096326888597709, −6.71836101468744863149723938997, −5.65268259531488216867176950815, −4.63061875593564800781886019952, −3.56552183985653330540012336856, −2.94130271851680950338012650731, −1.95853489662172806911323839925, −1.24163974682848041429704228794, 2.78309886840618212335624845726, 3.31693055882919299816924184871, 4.37809891029035343704705210648, 4.98935191078163767980275881796, 5.66034971747554470571603344086, 6.55744023294093180877093050785, 7.49412634701960054464873332878, 8.374166125700436649804920805743, 8.854404232624658576761278940108, 9.505022158150472488580609992090

Graph of the $Z$-function along the critical line