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Properties

Label 1-4729-4729.100-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.998 + 0.0613i$
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  + (0.198 + 0.980i)2-s + (−0.971 − 0.236i)3-s + (−0.921 + 0.388i)4-s + (−0.234 − 0.972i)5-s + (0.0398 − 0.999i)6-s + (−0.800 + 0.599i)7-s + (−0.563 − 0.826i)8-s + (0.887 + 0.460i)9-s + (0.906 − 0.422i)10-s + (0.991 + 0.127i)11-s + (0.987 − 0.158i)12-s + (−0.597 + 0.801i)13-s + (−0.746 − 0.665i)14-s + (−0.00265 + 0.999i)15-s + (0.698 − 0.715i)16-s + (−0.760 + 0.649i)17-s + ⋯
L(s)  = 1  + (0.198 + 0.980i)2-s + (−0.971 − 0.236i)3-s + (−0.921 + 0.388i)4-s + (−0.234 − 0.972i)5-s + (0.0398 − 0.999i)6-s + (−0.800 + 0.599i)7-s + (−0.563 − 0.826i)8-s + (0.887 + 0.460i)9-s + (0.906 − 0.422i)10-s + (0.991 + 0.127i)11-s + (0.987 − 0.158i)12-s + (−0.597 + 0.801i)13-s + (−0.746 − 0.665i)14-s + (−0.00265 + 0.999i)15-s + (0.698 − 0.715i)16-s + (−0.760 + 0.649i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.998 + 0.0613i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.998 + 0.0613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4274958568 + 0.01312271177i\)
\(L(\frac12)\) \(\approx\) \(0.4274958568 + 0.01312271177i\)
\(L(1)\) \(\approx\) \(0.5308380568 + 0.2195137477i\)
\(L(1)\) \(\approx\) \(0.5308380568 + 0.2195137477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.198 + 0.980i)T \)
3 \( 1 + (-0.971 - 0.236i)T \)
5 \( 1 + (-0.234 - 0.972i)T \)
7 \( 1 + (-0.800 + 0.599i)T \)
11 \( 1 + (0.991 + 0.127i)T \)
13 \( 1 + (-0.597 + 0.801i)T \)
17 \( 1 + (-0.760 + 0.649i)T \)
19 \( 1 + (-0.875 - 0.483i)T \)
23 \( 1 + (-0.998 - 0.0478i)T \)
29 \( 1 + (0.198 - 0.980i)T \)
31 \( 1 + (0.576 + 0.817i)T \)
37 \( 1 + (0.683 - 0.730i)T \)
41 \( 1 + (-0.793 + 0.608i)T \)
43 \( 1 + (0.952 + 0.303i)T \)
47 \( 1 + (0.877 - 0.479i)T \)
53 \( 1 + (-0.767 + 0.641i)T \)
59 \( 1 + (-0.998 + 0.0584i)T \)
61 \( 1 + (-0.462 - 0.886i)T \)
67 \( 1 + (-0.925 - 0.378i)T \)
71 \( 1 + (-0.140 + 0.990i)T \)
73 \( 1 + (0.361 - 0.932i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.809 + 0.586i)T \)
89 \( 1 + (-0.859 - 0.511i)T \)
97 \( 1 + (-0.800 + 0.599i)T \)
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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

−18.16040964643310579351490905817, −17.57788373408346415200462434712, −17.029191435595840401533491725534, −16.22567025803820064037347917101, −15.347871363002612279641409295, −14.82240477324680435703628717856, −13.93791702165881580475720779152, −13.4190212857435145043719787252, −12.324895472862491429057838059733, −12.22644325112383801319839814364, −11.21539099576872661967669076282, −10.83663708441310133344845942752, −10.08055317043777668372877048661, −9.79366518927581997695523510950, −8.88315360188706761075937111192, −7.735670376789191819374436289840, −6.925595688557340048608123321921, −6.23600075475494989222084818556, −5.748759416237701038731946854542, −4.495003869892782388918198264554, −4.160516687326478538558663081298, −3.35244513748831863828159222815, −2.65789037262796431816756006299, −1.60471672213764908705690905968, −0.5132944488830616136709019792, 0.24608138212324040002096704629, 1.41544188038730104814358627555, 2.41435017348598204237732680171, 3.94986529816313451029615249968, 4.303320641742127066777405873986, 4.9341517620218607961187550479, 5.91586576224154425175241846596, 6.32848614169029496661763477651, 6.83158253227194573316984986929, 7.778376104672749072069035666693, 8.53190915080621729320085094005, 9.30313453670568226485537724633, 9.62816405650072829998372052674, 10.73828210372407632070540031113, 11.903423291058422597839939329986, 12.118139953931368388954281837660, 12.74812454129417865519398589416, 13.37344330250454711291368032307, 14.100953711253029563036452045509, 15.12382352826117176198266843255, 15.65734615656207616086677435599, 16.18479667281617182103146288927, 17.00252950479125183001605106929, 17.093399012291406856109420764542, 17.86841880317241516499267648624

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