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Properties

Label 1-4009-4009.1037-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.972 + 0.232i$
Analytic cond. $18.6177$
Root an. cond. $18.6177$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.873 − 0.486i)2-s + (0.992 − 0.119i)3-s + (0.525 + 0.850i)4-s + (−0.599 + 0.800i)5-s + (−0.925 − 0.379i)6-s + (0.858 + 0.512i)7-s + (−0.0448 − 0.998i)8-s + (0.971 − 0.237i)9-s + (0.913 − 0.406i)10-s + (0.753 + 0.657i)11-s + (0.623 + 0.781i)12-s + (0.998 − 0.0598i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.447 + 0.894i)16-s + (−0.772 − 0.635i)17-s + ⋯
L(s)  = 1  + (−0.873 − 0.486i)2-s + (0.992 − 0.119i)3-s + (0.525 + 0.850i)4-s + (−0.599 + 0.800i)5-s + (−0.925 − 0.379i)6-s + (0.858 + 0.512i)7-s + (−0.0448 − 0.998i)8-s + (0.971 − 0.237i)9-s + (0.913 − 0.406i)10-s + (0.753 + 0.657i)11-s + (0.623 + 0.781i)12-s + (0.998 − 0.0598i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.447 + 0.894i)16-s + (−0.772 − 0.635i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.972 + 0.232i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.972 + 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.030340376 + 0.2387975640i\)
\(L(\frac12)\) \(\approx\) \(2.030340376 + 0.2387975640i\)
\(L(1)\) \(\approx\) \(1.171901428 + 0.0001598393381i\)
\(L(1)\) \(\approx\) \(1.171901428 + 0.0001598393381i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.873 - 0.486i)T \)
3 \( 1 + (0.992 - 0.119i)T \)
5 \( 1 + (-0.599 + 0.800i)T \)
7 \( 1 + (0.858 + 0.512i)T \)
11 \( 1 + (0.753 + 0.657i)T \)
13 \( 1 + (0.998 - 0.0598i)T \)
17 \( 1 + (-0.772 - 0.635i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (0.998 - 0.0598i)T \)
31 \( 1 + (0.623 + 0.781i)T \)
37 \( 1 + (0.753 - 0.657i)T \)
41 \( 1 + (0.251 + 0.967i)T \)
43 \( 1 + (0.365 - 0.930i)T \)
47 \( 1 + (-0.280 - 0.959i)T \)
53 \( 1 + (-0.999 - 0.0299i)T \)
59 \( 1 + (0.251 + 0.967i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (-0.733 - 0.680i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (0.887 - 0.460i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.575 - 0.817i)T \)
97 \( 1 + (-0.337 - 0.941i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.48997041963688673995045853551, −17.68486417603129013943408890063, −17.149264949526873595370530517101, −16.193995388446086917837021366924, −15.94552658721353152597914688606, −15.15675718540616132310018722268, −14.48838735336900557740050487705, −13.82510716319215342176697584571, −13.28783037888001668413670398768, −12.160641570175371243617563848275, −11.313884497375135591817077609582, −10.87781588662566362890632808001, −9.92148937792454427773777024453, −9.16476417698268781004173730147, −8.56846252651893862753659312775, −8.09463243973144490925704412857, −7.7196402897347238952304634109, −6.61737740198279940599153545060, −6.0041653933164252949773755489, −4.76767297501053168112032523312, −4.232966058097556430544103712980, −3.48949921988183070460035542213, −2.23942089642667033762769943919, −1.39025179597449051583754346109, −0.88638903964097417249142116951, 0.94493924929538586984040012027, 1.87384841053435591258101483481, 2.442772339268961630569471183047, 3.20119267766361884101845773940, 4.04402402766691653927579343583, 4.54839870233937671068525230327, 6.22574411239615744747754085375, 6.84795944111065522693666127715, 7.51142599996478985900907071595, 8.30137165863172470360330756184, 8.59267401864475212913779118647, 9.42834592463955560594928839799, 10.15952329812166989361148266618, 10.886945909941388502168519078, 11.626756858763272491944435505831, 12.0681540790129708362911917545, 12.90364041156291745358136309525, 13.93546139878731386393311250235, 14.36154824693825954382525204544, 15.31675652024351000377208844027, 15.632776326659860088037508447567, 16.34458922595815040750960271422, 17.63180403636284404134718246821, 18.0540843343362739355131597223, 18.41158931171477286201752526165

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