Properties

Label 1-6017-6017.5881-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: $\chi_{6017} (5881, \cdot )$
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

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

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