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Properties

Label 1-6017-6017.1073-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.686 + 0.727i$
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  + (−0.704 + 0.709i)2-s + (−0.309 − 0.951i)3-s + (−0.00805 − 0.999i)4-s + (−0.997 − 0.0643i)5-s + (0.892 + 0.450i)6-s + (0.541 + 0.840i)7-s + (0.715 + 0.698i)8-s + (−0.809 + 0.587i)9-s + (0.748 − 0.663i)10-s + (−0.948 + 0.316i)12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.247 + 0.968i)15-s + (−0.999 + 0.0161i)16-s + (−0.136 − 0.990i)17-s + (0.152 − 0.988i)18-s + ⋯
L(s)  = 1  + (−0.704 + 0.709i)2-s + (−0.309 − 0.951i)3-s + (−0.00805 − 0.999i)4-s + (−0.997 − 0.0643i)5-s + (0.892 + 0.450i)6-s + (0.541 + 0.840i)7-s + (0.715 + 0.698i)8-s + (−0.809 + 0.587i)9-s + (0.748 − 0.663i)10-s + (−0.948 + 0.316i)12-s + (−0.913 − 0.406i)13-s + (−0.978 − 0.207i)14-s + (0.247 + 0.968i)15-s + (−0.999 + 0.0161i)16-s + (−0.136 − 0.990i)17-s + (0.152 − 0.988i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.686 + 0.727i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1073, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.686 + 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6319796020 + 0.2724931775i\)
\(L(\frac12)\) \(\approx\) \(0.6319796020 + 0.2724931775i\)
\(L(1)\) \(\approx\) \(0.5796104626 + 0.03208463534i\)
\(L(1)\) \(\approx\) \(0.5796104626 + 0.03208463534i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.704 + 0.709i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.997 - 0.0643i)T \)
7 \( 1 + (0.541 + 0.840i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (-0.136 - 0.990i)T \)
19 \( 1 + (0.324 - 0.945i)T \)
23 \( 1 + (0.996 - 0.0804i)T \)
29 \( 1 + (0.607 + 0.794i)T \)
31 \( 1 + (0.681 + 0.732i)T \)
37 \( 1 + (0.384 - 0.923i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (-0.200 + 0.979i)T \)
47 \( 1 + (-0.726 + 0.686i)T \)
53 \( 1 + (0.619 + 0.784i)T \)
59 \( 1 + (0.324 + 0.945i)T \)
61 \( 1 + (-0.892 - 0.450i)T \)
67 \( 1 + (-0.200 + 0.979i)T \)
71 \( 1 + (-0.974 - 0.223i)T \)
73 \( 1 + (0.769 - 0.638i)T \)
79 \( 1 + (0.995 + 0.0965i)T \)
83 \( 1 + (-0.554 + 0.832i)T \)
89 \( 1 + (0.354 + 0.935i)T \)
97 \( 1 + (0.339 + 0.940i)T \)
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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

−17.42037856077805869271337850110, −16.93018781860003628591178326277, −16.638337283595568970601957161935, −15.80130749477658995796927879036, −15.05107601850244451913644554652, −14.589546376820085198188431722548, −13.648217754501510780990078311937, −12.782942071278743920465293962828, −11.928930609178642254869669863297, −11.59754221293340084620907913657, −11.013040173722433886591237231017, −10.208733576097817366673803782486, −10.00099451226783184404940205176, −9.03964498522568213171683089971, −8.24838702665138590434846266252, −7.87212763675142557215096970846, −7.04685902874017800718696308483, −6.26921785218274069716829896992, −4.91139595549612166147426761382, −4.513058896005126442778593201337, −3.82634258277831738210467173452, −3.31477466618165100859280240620, −2.363485748488515673156327713392, −1.26487349633222510954729204562, −0.389168531761596355744604370667, 0.69250648961893108887124623609, 1.31051470768007168960839119787, 2.59475795349346538675917522024, 2.83294710974488381820099703745, 4.6447033064485380288655977798, 4.96653333053402229778125018032, 5.635623923403392542319773303610, 6.61882575442334903084717449297, 7.199494883445560226318719634154, 7.611580122752714212214401555197, 8.32563400074305245711350049527, 8.91966773509147283142383001426, 9.49799101559981998888081753041, 10.86326945176039790426882004022, 11.013153816026152003070262641939, 11.97086564416712760531644141573, 12.280105051572603297087379576152, 13.21566897977285609871129482005, 14.04261670016601294699079412904, 14.727080583167379509028369493716, 15.18079135817930202120116342032, 16.01533549879487889770160641842, 16.39470011480990709731727427648, 17.36703315535073487088228949638, 17.86594597307130058438524441966

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