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Properties

Label 1-4729-4729.9-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.998 - 0.0511i$
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  + (0.675 − 0.737i)2-s + (−0.933 − 0.358i)3-s + (−0.0875 − 0.996i)4-s + (−0.481 − 0.876i)5-s + (−0.894 + 0.446i)6-s + (−0.182 − 0.983i)7-s + (−0.793 − 0.608i)8-s + (0.742 + 0.669i)9-s + (−0.971 − 0.236i)10-s + (−0.975 + 0.221i)11-s + (−0.275 + 0.961i)12-s + (0.949 + 0.313i)13-s + (−0.848 − 0.529i)14-s + (0.135 + 0.990i)15-s + (−0.984 + 0.174i)16-s + (−0.0717 − 0.997i)17-s + ⋯
L(s)  = 1  + (0.675 − 0.737i)2-s + (−0.933 − 0.358i)3-s + (−0.0875 − 0.996i)4-s + (−0.481 − 0.876i)5-s + (−0.894 + 0.446i)6-s + (−0.182 − 0.983i)7-s + (−0.793 − 0.608i)8-s + (0.742 + 0.669i)9-s + (−0.971 − 0.236i)10-s + (−0.975 + 0.221i)11-s + (−0.275 + 0.961i)12-s + (0.949 + 0.313i)13-s + (−0.848 − 0.529i)14-s + (0.135 + 0.990i)15-s + (−0.984 + 0.174i)16-s + (−0.0717 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.998 - 0.0511i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.998 - 0.0511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1422256844 + 0.003640747191i\)
\(L(\frac12)\) \(\approx\) \(-0.1422256844 + 0.003640747191i\)
\(L(1)\) \(\approx\) \(0.4205471127 - 0.6192566416i\)
\(L(1)\) \(\approx\) \(0.4205471127 - 0.6192566416i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.675 - 0.737i)T \)
3 \( 1 + (-0.933 - 0.358i)T \)
5 \( 1 + (-0.481 - 0.876i)T \)
7 \( 1 + (-0.182 - 0.983i)T \)
11 \( 1 + (-0.975 + 0.221i)T \)
13 \( 1 + (0.949 + 0.313i)T \)
17 \( 1 + (-0.0717 - 0.997i)T \)
19 \( 1 + (-0.812 + 0.582i)T \)
23 \( 1 + (-0.763 - 0.645i)T \)
29 \( 1 + (-0.675 - 0.737i)T \)
31 \( 1 + (-0.103 - 0.994i)T \)
37 \( 1 + (-0.614 - 0.788i)T \)
41 \( 1 + (-0.351 + 0.936i)T \)
43 \( 1 + (0.999 - 0.0159i)T \)
47 \( 1 + (-0.939 - 0.343i)T \)
53 \( 1 + (-0.576 - 0.817i)T \)
59 \( 1 + (0.987 - 0.158i)T \)
61 \( 1 + (0.563 + 0.826i)T \)
67 \( 1 + (-0.589 + 0.808i)T \)
71 \( 1 + (0.783 + 0.620i)T \)
73 \( 1 + (0.00797 - 0.999i)T \)
79 \( 1 - T \)
83 \( 1 + (0.839 - 0.543i)T \)
89 \( 1 + (-0.627 - 0.778i)T \)
97 \( 1 + (-0.182 - 0.983i)T \)
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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

−18.64433397716722265920764040811, −17.86621667406551943603717677591, −17.65805291536971379418816960750, −16.58857179970364837875692842917, −15.862234674409714557107160311924, −15.53835969683390784176389322321, −15.15838469166879114924398683332, −14.3304658142180772020784848923, −13.41383047561969834405802140153, −12.6716781531476677241617511472, −12.2856315660683403535447823591, −11.35180666004933292538765053875, −10.91435772287121680013250832464, −10.27269067870910428138107941437, −9.145129161089822810268246990802, −8.38255058733854426369772460641, −7.7902916275281865438136985637, −6.73975863087897744430525496842, −6.39773465469553283224664979612, −5.5977960992613818983410717912, −5.22009319185994322708889665523, −4.14439235638211513428905396609, −3.53831007795980159220394464724, −2.87227759279594577639587050268, −1.81709310394932819882758474390, 0.04955956008861419622630157914, 0.667618758636794119580978767892, 1.5987591098283144588881607571, 2.32347092762531482042319933753, 3.633591546130977233769152069905, 4.22582799553299077546010153982, 4.71218544505223915915559645653, 5.55668046955929540113361726610, 6.13680580933030934700843796638, 7.00272008807635788903184700243, 7.753002235384872831558624340574, 8.533141962132306756554268064305, 9.68973266453864591671432503999, 10.15066178981817068609159695063, 10.99010948309648287493850208750, 11.4009545964736011341637059961, 12.0812699805022972563683728991, 12.97635355016968162841642746296, 13.0765454355562623374356735211, 13.765991313064643073539665391392, 14.67664969120451665230563042244, 15.7162041086547817257411828441, 16.14973701202253185204765506341, 16.63537283001143431321172163932, 17.58928265892779319371503895746

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