Properties

Label 1-6017-6017.1460-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.961 - 0.275i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
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  + (−0.993 − 0.112i)2-s + (0.809 − 0.587i)3-s + (0.974 + 0.223i)4-s + (0.231 + 0.972i)5-s + (−0.870 + 0.493i)6-s + (−0.953 − 0.301i)7-s + (−0.943 − 0.331i)8-s + (0.309 − 0.951i)9-s + (−0.120 − 0.992i)10-s + (0.919 − 0.391i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (0.759 + 0.650i)15-s + (0.899 + 0.435i)16-s + (−0.769 − 0.638i)17-s + (−0.414 + 0.910i)18-s + ⋯
L(s)  = 1  + (−0.993 − 0.112i)2-s + (0.809 − 0.587i)3-s + (0.974 + 0.223i)4-s + (0.231 + 0.972i)5-s + (−0.870 + 0.493i)6-s + (−0.953 − 0.301i)7-s + (−0.943 − 0.331i)8-s + (0.309 − 0.951i)9-s + (−0.120 − 0.992i)10-s + (0.919 − 0.391i)12-s + (−0.669 − 0.743i)13-s + (0.913 + 0.406i)14-s + (0.759 + 0.650i)15-s + (0.899 + 0.435i)16-s + (−0.769 − 0.638i)17-s + (−0.414 + 0.910i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.961 - 0.275i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.961 - 0.275i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09740147166 - 0.6922194437i\)
\(L(\frac12)\) \(\approx\) \(0.09740147166 - 0.6922194437i\)
\(L(1)\) \(\approx\) \(0.7251205607 - 0.2314018232i\)
\(L(1)\) \(\approx\) \(0.7251205607 - 0.2314018232i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.993 - 0.112i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.231 + 0.972i)T \)
7 \( 1 + (-0.953 - 0.301i)T \)
13 \( 1 + (-0.669 - 0.743i)T \)
17 \( 1 + (-0.769 - 0.638i)T \)
19 \( 1 + (0.984 + 0.176i)T \)
23 \( 1 + (0.632 - 0.774i)T \)
29 \( 1 + (-0.836 + 0.548i)T \)
31 \( 1 + (0.527 - 0.849i)T \)
37 \( 1 + (-0.0563 - 0.998i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (0.799 + 0.600i)T \)
47 \( 1 + (-0.704 + 0.709i)T \)
53 \( 1 + (0.991 - 0.128i)T \)
59 \( 1 + (0.984 - 0.176i)T \)
61 \( 1 + (0.870 - 0.493i)T \)
67 \( 1 + (0.799 + 0.600i)T \)
71 \( 1 + (-0.999 + 0.0322i)T \)
73 \( 1 + (-0.853 - 0.520i)T \)
79 \( 1 + (-0.906 - 0.421i)T \)
83 \( 1 + (-0.726 + 0.686i)T \)
89 \( 1 + (0.748 + 0.663i)T \)
97 \( 1 + (-0.962 - 0.270i)T \)
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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

−17.934857701800745435816379135280, −17.19069630034510395282717725468, −16.655916329765047975531965197295, −16.11981510147650033702049139673, −15.540776830583367612578150095810, −15.06668120973553317458243329422, −14.1738795659445978209407768025, −13.317087509265452160408244764362, −12.91256347956395526950201485956, −11.86001132450025900227228213855, −11.444989036993499963032033200743, −10.24628191295603670842318446521, −9.86365240741803799931839279581, −9.30814128259193239932192426687, −8.80422617436094393742678242420, −8.31079240615917374987683609461, −7.34691643509647880853897934569, −6.82950366897419316783072944682, −5.81280459971394807327972966862, −5.179131255455612341447601942063, −4.29448570985811428206616343515, −3.41535806932620252735231306327, −2.647641525747596267854768508304, −1.94872679522671714370215717509, −1.15159690322739467430802241462, 0.229584607194405246242131725226, 1.09480144018938251352602798938, 2.33676518050262537663340087596, 2.58628653643152939511452843873, 3.28101144970196160147404434228, 3.9605014808657792681758982892, 5.499720349277951464370810397156, 6.26491059321581833440198594624, 7.02283293344606704264394848970, 7.25321378155033042645688735436, 7.900522784858591962244384893776, 8.884069070195264795540410479310, 9.44029077861885590590082276922, 9.947526931974690828169866011132, 10.58164838774729413046693614816, 11.36354985977139173168584343256, 12.088605055132760303706303090161, 12.95463935332755234591145823431, 13.27308242085783612785452504109, 14.40409774089800822723360965182, 14.66412531110174622976817984777, 15.65789595690034452161753531927, 15.93696696853154450381320784892, 16.98748738885775478487198662229, 17.64765939559487219612029594882

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