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Properties

Label 1-4729-4729.413-r0-0-0
Degree $1$
Conductor $4729$
Sign $-0.805 + 0.592i$
Analytic cond. $21.9613$
Root an. cond. $21.9613$
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  + (0.698 + 0.715i)2-s + (0.576 − 0.817i)3-s + (−0.0239 + 0.999i)4-s + (0.410 − 0.912i)5-s + (0.987 − 0.158i)6-s + (0.887 + 0.460i)7-s + (−0.732 + 0.681i)8-s + (−0.336 − 0.941i)9-s + (0.939 − 0.343i)10-s + (−0.872 + 0.488i)11-s + (0.803 + 0.595i)12-s + (−0.0557 + 0.998i)13-s + (0.290 + 0.956i)14-s + (−0.509 − 0.860i)15-s + (−0.998 − 0.0478i)16-s + (−0.742 − 0.669i)17-s + ⋯
L(s)  = 1  + (0.698 + 0.715i)2-s + (0.576 − 0.817i)3-s + (−0.0239 + 0.999i)4-s + (0.410 − 0.912i)5-s + (0.987 − 0.158i)6-s + (0.887 + 0.460i)7-s + (−0.732 + 0.681i)8-s + (−0.336 − 0.941i)9-s + (0.939 − 0.343i)10-s + (−0.872 + 0.488i)11-s + (0.803 + 0.595i)12-s + (−0.0557 + 0.998i)13-s + (0.290 + 0.956i)14-s + (−0.509 − 0.860i)15-s + (−0.998 − 0.0478i)16-s + (−0.742 − 0.669i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $-0.805 + 0.592i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ -0.805 + 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5085674108 + 1.548556799i\)
\(L(\frac12)\) \(\approx\) \(0.5085674108 + 1.548556799i\)
\(L(1)\) \(\approx\) \(1.495205903 + 0.4253137875i\)
\(L(1)\) \(\approx\) \(1.495205903 + 0.4253137875i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.698 + 0.715i)T \)
3 \( 1 + (0.576 - 0.817i)T \)
5 \( 1 + (0.410 - 0.912i)T \)
7 \( 1 + (0.887 + 0.460i)T \)
11 \( 1 + (-0.872 + 0.488i)T \)
13 \( 1 + (-0.0557 + 0.998i)T \)
17 \( 1 + (-0.742 - 0.669i)T \)
19 \( 1 + (-0.563 + 0.826i)T \)
23 \( 1 + (-0.981 + 0.190i)T \)
29 \( 1 + (-0.698 + 0.715i)T \)
31 \( 1 + (0.773 - 0.633i)T \)
37 \( 1 + (0.380 + 0.924i)T \)
41 \( 1 + (0.864 - 0.502i)T \)
43 \( 1 + (-0.651 - 0.758i)T \)
47 \( 1 + (-0.995 + 0.0955i)T \)
53 \( 1 + (-0.166 + 0.986i)T \)
59 \( 1 + (-0.687 + 0.726i)T \)
61 \( 1 + (0.639 + 0.768i)T \)
67 \( 1 + (0.856 + 0.516i)T \)
71 \( 1 + (0.0398 + 0.999i)T \)
73 \( 1 + (-0.908 + 0.417i)T \)
79 \( 1 - T \)
83 \( 1 + (0.915 - 0.402i)T \)
89 \( 1 + (0.453 - 0.891i)T \)
97 \( 1 + (0.887 + 0.460i)T \)
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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

−17.8393481504394019715122854704, −17.55546586494629660545996594849, −16.30187675513686438741647219601, −15.49803557663506469003410662106, −15.07743056981959976505313883887, −14.50742401834541547431950460571, −13.90366082562583175493511242115, −13.28857882055895188659637771044, −12.80267515336011502804834881462, −11.32221658546669376500319212926, −11.15571190136890550576306570307, −10.45883187608643655181407446881, −10.09661224083034278042154545050, −9.27000315182331230597877467173, −8.220039440372118518165016235135, −7.82384755553782197297119685, −6.60898708927098943675312406016, −5.87810588875708700362869548812, −5.11367838874714429537037233031, −4.499633440076895005883151947360, −3.71908537242935858597603966050, −3.02760054772216505119401354030, −2.34392644542775351579779563523, −1.80190704564155954870233400981, −0.25604783672046482898840974690, 1.413895421428466160103787174806, 2.172864667292636645010062712600, 2.577946756625824562605892099781, 3.94710659458643436249058132747, 4.50447678862356252716838578393, 5.25585869644231015532637070354, 5.943751931104136459573730337656, 6.63036306763034225277402513699, 7.572712692487268941710580209779, 7.96313425667246917289476654948, 8.73617020558668470870118352425, 9.11400305721806921856615114569, 10.11111683216161040766467819717, 11.52822739478749651241466908649, 11.85449048976127236968647009937, 12.5897376818709694952556315407, 13.17409110344574175413730645990, 13.72077254158804168710984719873, 14.35083680897290719664494010687, 14.90648148450142869886611470972, 15.70552857302432644321112128551, 16.28190979391147177265790718736, 17.26077216197363484460204144828, 17.54559017820370882519066968161, 18.439451667039214236384455505909

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