↑

Properties

Label 1-4009-4009.2100-r1-0-0
Degree $1$
Conductor $4009$
Sign $0.794 + 0.607i$
Analytic cond. $430.826$
Root an. cond. $430.826$
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  + (0.882 + 0.469i)2-s + (0.997 + 0.0697i)3-s + (0.559 + 0.829i)4-s + (−0.241 − 0.970i)5-s + (0.848 + 0.529i)6-s + (0.309 + 0.951i)7-s + (0.104 + 0.994i)8-s + (0.990 + 0.139i)9-s + (0.241 − 0.970i)10-s + (0.913 − 0.406i)11-s + (0.5 + 0.866i)12-s + (0.882 − 0.469i)13-s + (−0.173 + 0.984i)14-s + (−0.173 − 0.984i)15-s + (−0.374 + 0.927i)16-s + (0.990 − 0.139i)17-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)2-s + (0.997 + 0.0697i)3-s + (0.559 + 0.829i)4-s + (−0.241 − 0.970i)5-s + (0.848 + 0.529i)6-s + (0.309 + 0.951i)7-s + (0.104 + 0.994i)8-s + (0.990 + 0.139i)9-s + (0.241 − 0.970i)10-s + (0.913 − 0.406i)11-s + (0.5 + 0.866i)12-s + (0.882 − 0.469i)13-s + (−0.173 + 0.984i)14-s + (−0.173 − 0.984i)15-s + (−0.374 + 0.927i)16-s + (0.990 − 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(430.826\)
Root analytic conductor: \(430.826\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (2100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (1:\ ),\ 0.794 + 0.607i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(8.491047213 + 2.873199173i\)
\(L(\frac12)\) \(\approx\) \(8.491047213 + 2.873199173i\)
\(L(1)\) \(\approx\) \(2.922064013 + 0.7697238707i\)
\(L(1)\) \(\approx\) \(2.922064013 + 0.7697238707i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.882 + 0.469i)T \)
3 \( 1 + (0.997 + 0.0697i)T \)
5 \( 1 + (-0.241 - 0.970i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.882 - 0.469i)T \)
17 \( 1 + (0.990 - 0.139i)T \)
23 \( 1 + (-0.374 - 0.927i)T \)
29 \( 1 + (0.882 - 0.469i)T \)
31 \( 1 - T \)
37 \( 1 + (0.104 + 0.994i)T \)
41 \( 1 + (-0.961 - 0.275i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.848 + 0.529i)T \)
53 \( 1 + (0.719 + 0.694i)T \)
59 \( 1 + (0.719 - 0.694i)T \)
61 \( 1 + (0.961 - 0.275i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.997 + 0.0697i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.719 - 0.694i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.882 - 0.469i)T \)
97 \( 1 + (-0.559 + 0.829i)T \)
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

−18.39294480519366033392120802971, −17.87683727988571278332757588909, −16.63131827941755421394782629103, −15.98946618009621362803446820863, −15.166171734985185016845455194855, −14.55326124142731507574950285644, −14.174134990501395404602675325774, −13.699888481415205336813236396359, −12.938721697087926839189851015152, −12.05377744290925637275742383379, −11.414016726915396816921346434, −10.70535556125508665119332889873, −10.03556086268518326250311636075, −9.48851250310920393526635079073, −8.416250223584903564339395920393, −7.49832633010653630480348272376, −6.99828316344385392748760325324, −6.42903238038242568137143713504, −5.38183426847224025196705415640, −4.241409627707364207892507657440, −3.71843725936611514582160778763, −3.46315582665979349732432883699, −2.36091770018370618340918054378, −1.58778032761866817379603610058, −0.96200563532031225870237552697, 0.90488349853423618914792128237, 1.77025614682102043765362277036, 2.63100277061620896688739526066, 3.52544764153845366042100871315, 3.991228983531804474252320412468, 4.86804261485047016909319144553, 5.5593134995554420120454307836, 6.26242693033087253937262877415, 7.20146577075534609123492558619, 8.18289169179236296717516926809, 8.42925563490337235509030048713, 8.96801375427639497675531929675, 9.913497800814511187898808366344, 11.00048073222560351346534648127, 11.98141519039146996324588726596, 12.2362937999199344646281596858, 13.02707398113947344072165295708, 13.69784611079147784014290226993, 14.32100694631296929161777586134, 14.90435135386884914845238215391, 15.665422120370427881905297418582, 16.03324915907903384273911411316, 16.76505325362755031794792692475, 17.51462290630763166710894846284, 18.56593841766415068096669805174

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