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Properties

Label 1-4033-4033.1053-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.553 + 0.832i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (−0.5 + 0.866i)2-s + (−0.973 + 0.230i)3-s + (−0.5 − 0.866i)4-s + (0.998 − 0.0581i)5-s + (0.286 − 0.957i)6-s + (0.893 + 0.448i)7-s + 8-s + (0.893 − 0.448i)9-s + (−0.448 + 0.893i)10-s + (0.448 + 0.893i)11-s + (0.686 + 0.727i)12-s + (−0.0581 − 0.998i)13-s + (−0.835 + 0.549i)14-s + (−0.957 + 0.286i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.973 + 0.230i)3-s + (−0.5 − 0.866i)4-s + (0.998 − 0.0581i)5-s + (0.286 − 0.957i)6-s + (0.893 + 0.448i)7-s + 8-s + (0.893 − 0.448i)9-s + (−0.448 + 0.893i)10-s + (0.448 + 0.893i)11-s + (0.686 + 0.727i)12-s + (−0.0581 − 0.998i)13-s + (−0.835 + 0.549i)14-s + (−0.957 + 0.286i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.297577897 + 0.6956330873i\)
\(L(\frac12)\) \(\approx\) \(1.297577897 + 0.6956330873i\)
\(L(1)\) \(\approx\) \(0.8266703599 + 0.3579282474i\)
\(L(1)\) \(\approx\) \(0.8266703599 + 0.3579282474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.973 + 0.230i)T \)
5 \( 1 + (0.998 - 0.0581i)T \)
7 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (0.448 + 0.893i)T \)
13 \( 1 + (-0.0581 - 0.998i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.549 - 0.835i)T \)
31 \( 1 + (-0.802 + 0.597i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (0.957 - 0.286i)T \)
53 \( 1 + (-0.957 - 0.286i)T \)
59 \( 1 + (-0.286 - 0.957i)T \)
61 \( 1 + (0.802 + 0.597i)T \)
67 \( 1 + (-0.727 - 0.686i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.835 + 0.549i)T \)
79 \( 1 + (0.973 + 0.230i)T \)
83 \( 1 + (-0.597 + 0.802i)T \)
89 \( 1 + (0.448 - 0.893i)T \)
97 \( 1 + (0.802 + 0.597i)T \)
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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

−18.44755829724672080529350522758, −17.77661019420379409801187628365, −17.016144349172011684484616600607, −16.67340513140726512792840341357, −16.23449759576624149691336952632, −14.63484242023199446079631613690, −14.05159872941314557851197324208, −13.51160463769584783386911885802, −12.75139950110783279351995687607, −11.98346572697367753761919436035, −11.28328914997234344971611354575, −10.98535138384538913100947913661, −10.22881065569500143994202503826, −9.40008874264220515018078266301, −8.98075101207987234324370936420, −7.79966282344024972874111438458, −7.28199392002002686988750967196, −6.39950988227274085233292230584, −5.51471871893858378185017536634, −4.90630696959593759976770345002, −4.0817986274076752997830813232, −3.17159684137842027291162001636, −2.03178808889588526589573004637, −1.42575008121418720365408912287, −0.87466026108056698006542391676, 0.78631977782461798681950968230, 1.53011215747997599364866789557, 2.32831772523066252014126347992, 3.87878154873705006394440426603, 4.8957371437093418581349696424, 5.21082602460281675776681169243, 5.83114334158180185388710943530, 6.58262517656838799387177600082, 7.25025966601818841744774933417, 8.043555977672773655661223851825, 9.040282053494201261688108265095, 9.44008475930618735382802599293, 10.345337132225763754440764306624, 10.71330561088370021871132121091, 11.53224988560892450624514665248, 12.63525310828641317946264326511, 12.95569289221600673972763515785, 14.03679686705406915264625383577, 14.80358859186936655064177015122, 15.18007534267429792497826025561, 15.85413279634690200497854694656, 16.93517278623061676871373685984, 17.2650436821430904143461589889, 17.60540503732081930199934617977, 18.27081924313442189947282325403

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