↑

Properties

Label 1-6001-6001.1629-r1-0-0
Degree $1$
Conductor $6001$
Sign $0.390 + 0.920i$
Analytic cond. $644.897$
Root an. cond. $644.897$
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  + (−0.479 + 0.877i)2-s + (0.778 + 0.627i)3-s + (−0.540 − 0.841i)4-s + (−0.627 + 0.778i)5-s + (−0.923 + 0.382i)6-s + (0.382 + 0.923i)7-s + (0.997 − 0.0713i)8-s + (0.212 + 0.977i)9-s + (−0.382 − 0.923i)10-s + (−0.510 − 0.860i)11-s + (0.106 − 0.994i)12-s + (0.909 + 0.415i)13-s + (−0.994 − 0.106i)14-s + (−0.977 + 0.212i)15-s + (−0.415 + 0.909i)16-s + ⋯
L(s)  = 1  + (−0.479 + 0.877i)2-s + (0.778 + 0.627i)3-s + (−0.540 − 0.841i)4-s + (−0.627 + 0.778i)5-s + (−0.923 + 0.382i)6-s + (0.382 + 0.923i)7-s + (0.997 − 0.0713i)8-s + (0.212 + 0.977i)9-s + (−0.382 − 0.923i)10-s + (−0.510 − 0.860i)11-s + (0.106 − 0.994i)12-s + (0.909 + 0.415i)13-s + (−0.994 − 0.106i)14-s + (−0.977 + 0.212i)15-s + (−0.415 + 0.909i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.390 + 0.920i$
Analytic conductor: \(644.897\)
Root analytic conductor: \(644.897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (1629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (1:\ ),\ 0.390 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.466643990 + 0.9713357451i\)
\(L(\frac12)\) \(\approx\) \(1.466643990 + 0.9713357451i\)
\(L(1)\) \(\approx\) \(0.7343958779 + 0.6545359134i\)
\(L(1)\) \(\approx\) \(0.7343958779 + 0.6545359134i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.479 + 0.877i)T \)
3 \( 1 + (0.778 + 0.627i)T \)
5 \( 1 + (-0.627 + 0.778i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (-0.510 - 0.860i)T \)
13 \( 1 + (0.909 + 0.415i)T \)
19 \( 1 + (0.936 + 0.349i)T \)
23 \( 1 + (-0.247 - 0.968i)T \)
29 \( 1 + (-0.821 + 0.570i)T \)
31 \( 1 + (0.247 - 0.968i)T \)
37 \( 1 + (0.894 - 0.447i)T \)
41 \( 1 + (0.860 - 0.510i)T \)
43 \( 1 + (0.599 - 0.800i)T \)
47 \( 1 + (0.540 + 0.841i)T \)
53 \( 1 + (-0.877 + 0.479i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.999 - 0.0356i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.984 + 0.177i)T \)
73 \( 1 + (-0.177 - 0.984i)T \)
79 \( 1 + (-0.778 + 0.627i)T \)
83 \( 1 + (0.997 + 0.0713i)T \)
89 \( 1 + (-0.755 + 0.654i)T \)
97 \( 1 + (0.948 - 0.315i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.76499121025838648882897163229, −17.00577116898246613945510935563, −16.12507505065082928373861119401, −15.5794254276445619773582508814, −14.71166266246404485988892210930, −13.80427011694352003330450670222, −13.277190457901128298068717095991, −12.95463609467631322525862022456, −12.12441644423789212795680786159, −11.55197264598291667866495304790, −10.90103264025879053220742072508, −9.9899650201865387515152891721, −9.429837300774973323448743277828, −8.75292846659932961791697113734, −7.99817215030650275835247857888, −7.573743151073067910059214769116, −7.23550466669440580965725255204, −5.8886119725080578151769431908, −4.7541948352741148782267129283, −4.2665011706701670743264948533, −3.47512234098882995615560287180, −2.93845607958846698460474343577, −1.80751446007881599073575414746, −1.243570247403405158428713061239, −0.71253541505557144562105612425, 0.3327341467443366931414551683, 1.55310458277196348948131825708, 2.49314299731656946466728060481, 3.188776735755725225896327024657, 4.06766909313883693170902006713, 4.64801353769077220156784435925, 5.84386183762144775396782147623, 5.90834527016384759957910523089, 7.13265358459442532099477422176, 7.95180344840267020631769079671, 8.05526216119185942634502841850, 9.1299144694302621949080205972, 9.21918804099083564412723501833, 10.40126693379116851791577308189, 10.89644645474417946835703878225, 11.37605010107283806247465728835, 12.480630636060608364617695601087, 13.469116903703055847679111364399, 14.1198597663722748357557254838, 14.47498175798452446074976715195, 15.21520223897287465786512012204, 15.70438124991184751932061270733, 16.16316085184824530824146606622, 16.67905682232375849894479776302, 17.88664117664852731207496497299

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