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Properties

Label 1-4729-4729.3408-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.828 - 0.560i$
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  + (0.351 + 0.936i)2-s + (−0.908 − 0.417i)3-s + (−0.753 + 0.657i)4-s + (0.593 + 0.804i)5-s + (0.0717 − 0.997i)6-s + (−0.739 + 0.673i)7-s + (−0.880 − 0.474i)8-s + (0.651 + 0.758i)9-s + (−0.545 + 0.838i)10-s + (0.921 − 0.388i)11-s + (0.959 − 0.283i)12-s + (0.809 − 0.586i)13-s + (−0.890 − 0.455i)14-s + (−0.203 − 0.979i)15-s + (0.135 − 0.990i)16-s + (−0.513 − 0.857i)17-s + ⋯
L(s)  = 1  + (0.351 + 0.936i)2-s + (−0.908 − 0.417i)3-s + (−0.753 + 0.657i)4-s + (0.593 + 0.804i)5-s + (0.0717 − 0.997i)6-s + (−0.739 + 0.673i)7-s + (−0.880 − 0.474i)8-s + (0.651 + 0.758i)9-s + (−0.545 + 0.838i)10-s + (0.921 − 0.388i)11-s + (0.959 − 0.283i)12-s + (0.809 − 0.586i)13-s + (−0.890 − 0.455i)14-s + (−0.203 − 0.979i)15-s + (0.135 − 0.990i)16-s + (−0.513 − 0.857i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.828 - 0.560i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (3408, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.828 - 0.560i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4957697924 - 0.1520311543i\)
\(L(\frac12)\) \(\approx\) \(0.4957697924 - 0.1520311543i\)
\(L(1)\) \(\approx\) \(0.7025423158 + 0.4062630952i\)
\(L(1)\) \(\approx\) \(0.7025423158 + 0.4062630952i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.351 + 0.936i)T \)
3 \( 1 + (-0.908 - 0.417i)T \)
5 \( 1 + (0.593 + 0.804i)T \)
7 \( 1 + (-0.739 + 0.673i)T \)
11 \( 1 + (0.921 - 0.388i)T \)
13 \( 1 + (0.809 - 0.586i)T \)
17 \( 1 + (-0.513 - 0.857i)T \)
19 \( 1 + (-0.375 + 0.926i)T \)
23 \( 1 + (-0.856 + 0.516i)T \)
29 \( 1 + (-0.351 + 0.936i)T \)
31 \( 1 + (0.150 - 0.988i)T \)
37 \( 1 + (0.979 + 0.200i)T \)
41 \( 1 + (0.996 + 0.0796i)T \)
43 \( 1 + (-0.177 - 0.984i)T \)
47 \( 1 + (-0.713 - 0.700i)T \)
53 \( 1 + (-0.977 - 0.211i)T \)
59 \( 1 + (0.208 + 0.978i)T \)
61 \( 1 + (-0.610 - 0.792i)T \)
67 \( 1 + (-0.0345 - 0.999i)T \)
71 \( 1 + (0.988 + 0.148i)T \)
73 \( 1 + (-0.171 - 0.985i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.385 + 0.922i)T \)
89 \( 1 + (-0.877 + 0.479i)T \)
97 \( 1 + (-0.739 + 0.673i)T \)
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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

−18.06306636398796244427551089898, −17.49740327228438397831307372735, −17.08327280062460748260776041598, −16.242773386010885518619773672499, −15.74593480845804247923406330375, −14.70928426032527728690040336812, −14.02772567424982355854579602827, −13.2145470929163154834463505845, −12.799645991108549298492075412184, −12.2357557205844318323776596761, −11.33814745048100943766789880325, −10.92388899650255328003035333025, −10.0899200149921485780501936364, −9.517305603140231633571985801019, −9.127854126261823044177537620495, −8.20820121148183694621733527099, −6.65165795511059971845947423441, −6.334930327037866623378229990136, −5.724814478997989889244308383838, −4.54637670320882859301670934785, −4.31921283274337726750461528234, −3.74374607360686602511230324816, −2.52586335655480550087335658788, −1.48760993239914132931050941592, −1.00305968250768156753495803553, 0.159543640614492063090004002309, 1.48572004623824285983324896819, 2.524122121833999386545778921147, 3.44101945978669825077490693882, 4.08479720301620435190036024872, 5.28884563516357328677752378083, 5.79678758156535609375289155800, 6.341903663222711915487134002837, 6.66145408843927955290091507393, 7.57275742991226636923924671862, 8.29632125312798728898850133787, 9.33521314676283279863525421525, 9.718428480439768237696047173974, 10.74854740448923570674916329707, 11.44614048425211438535879811939, 12.16385402729246740875284836068, 12.80820854267331748681765962049, 13.56033749672441269154205128756, 13.89596515156944556790638027285, 14.893987798994329676110729392783, 15.40928623204657666053794697906, 16.35272228642221640901404869120, 16.50626825046635656657370461950, 17.47620194268913046109124523961, 18.023085006543691672318845898698

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