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Properties

Label 1-4729-4729.1047-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.587 + 0.809i$
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  + (−0.753 + 0.657i)2-s + (0.651 + 0.758i)3-s + (0.135 − 0.990i)4-s + (−0.679 − 0.733i)5-s + (−0.989 − 0.143i)6-s + (0.815 + 0.578i)7-s + (0.549 + 0.835i)8-s + (−0.150 + 0.988i)9-s + (0.994 + 0.106i)10-s + (0.698 − 0.715i)11-s + (0.839 − 0.543i)12-s + (0.667 + 0.744i)13-s + (−0.994 + 0.100i)14-s + (0.114 − 0.993i)15-s + (−0.963 − 0.267i)16-s + (−0.527 − 0.849i)17-s + ⋯
L(s)  = 1  + (−0.753 + 0.657i)2-s + (0.651 + 0.758i)3-s + (0.135 − 0.990i)4-s + (−0.679 − 0.733i)5-s + (−0.989 − 0.143i)6-s + (0.815 + 0.578i)7-s + (0.549 + 0.835i)8-s + (−0.150 + 0.988i)9-s + (0.994 + 0.106i)10-s + (0.698 − 0.715i)11-s + (0.839 − 0.543i)12-s + (0.667 + 0.744i)13-s + (−0.994 + 0.100i)14-s + (0.114 − 0.993i)15-s + (−0.963 − 0.267i)16-s + (−0.527 − 0.849i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.587 + 0.809i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1047, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.587 + 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.457328635 + 0.7428383259i\)
\(L(\frac12)\) \(\approx\) \(1.457328635 + 0.7428383259i\)
\(L(1)\) \(\approx\) \(0.9212069425 + 0.3767579359i\)
\(L(1)\) \(\approx\) \(0.9212069425 + 0.3767579359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.753 + 0.657i)T \)
3 \( 1 + (0.651 + 0.758i)T \)
5 \( 1 + (-0.679 - 0.733i)T \)
7 \( 1 + (0.815 + 0.578i)T \)
11 \( 1 + (0.698 - 0.715i)T \)
13 \( 1 + (0.667 + 0.744i)T \)
17 \( 1 + (-0.527 - 0.849i)T \)
19 \( 1 + (0.962 - 0.272i)T \)
23 \( 1 + (0.467 - 0.884i)T \)
29 \( 1 + (-0.753 - 0.657i)T \)
31 \( 1 + (-0.954 - 0.298i)T \)
37 \( 1 + (-0.800 + 0.599i)T \)
41 \( 1 + (0.987 + 0.158i)T \)
43 \( 1 + (0.770 + 0.637i)T \)
47 \( 1 + (-0.875 - 0.483i)T \)
53 \( 1 + (-0.0981 - 0.995i)T \)
59 \( 1 + (0.809 + 0.586i)T \)
61 \( 1 + (0.964 - 0.262i)T \)
67 \( 1 + (0.558 + 0.829i)T \)
71 \( 1 + (-0.224 - 0.974i)T \)
73 \( 1 + (0.177 - 0.984i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.265 + 0.964i)T \)
89 \( 1 + (0.458 + 0.888i)T \)
97 \( 1 + (0.815 + 0.578i)T \)
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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

−18.07929631744238820935828460090, −17.63988998649624288427228961928, −17.1891210352897651423597137905, −15.97396625135183609568053325275, −15.411403970068911537672698878903, −14.50139609003871760348180330538, −14.18015715158782222565466705373, −13.0999698441818780467565081580, −12.68385861502853989846672637493, −11.82460310019991667404214372070, −11.24591309710226162433413313611, −10.74393066018060304925794067575, −9.92099682241446851920145068109, −9.026950741585283778051889430154, −8.50188844902846053571490433617, −7.611330498057910489247171451937, −7.400776719578927781354721380027, −6.79491674025401763278062426091, −5.68717410286077319132436650206, −4.310146115620995889453027249286, −3.61345073147271240359912683653, −3.294745822878390546579796052191, −2.11374306683056327746866851196, −1.56877757871403430176542220698, −0.79268493845302194171445556094, 0.73051803487634970153330866450, 1.63992416686932568240852831417, 2.49251993837529209339829502526, 3.62629884175491556296502288303, 4.37978558773266603827858935366, 5.05141215597480885563186887982, 5.6092183175608551224607657017, 6.67494573504227536414229833819, 7.520273862032176365201757716119, 8.19660960374208805394185664228, 8.75094003518249415339681593944, 9.158679675479515157118133656256, 9.65805218946288196080873135890, 10.95380890015949323235219714706, 11.3139466698789366453566820750, 11.77750520413176450173610562227, 13.152609188124446940016779358219, 13.8306916706720307126907279991, 14.52370707952849807384722491752, 14.967646317855911754764226761561, 15.84187754947216374367102494394, 16.166812601888089028798743051219, 16.656359474125352221219673040598, 17.47219868480362744595824554649, 18.33811656871500926852110824495

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