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Properties

Label 1-6017-6017.922-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.0307 + 0.999i$
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  + (−0.681 + 0.732i)2-s + (0.309 − 0.951i)3-s + (−0.0724 − 0.997i)4-s + (0.836 + 0.548i)5-s + (0.485 + 0.873i)6-s + (−0.906 + 0.421i)7-s + (0.779 + 0.626i)8-s + (−0.809 − 0.587i)9-s + (−0.970 + 0.239i)10-s + (−0.970 − 0.239i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (0.779 − 0.626i)15-s + (−0.989 + 0.144i)16-s + (−0.943 − 0.331i)17-s + (0.981 − 0.192i)18-s + ⋯
L(s)  = 1  + (−0.681 + 0.732i)2-s + (0.309 − 0.951i)3-s + (−0.0724 − 0.997i)4-s + (0.836 + 0.548i)5-s + (0.485 + 0.873i)6-s + (−0.906 + 0.421i)7-s + (0.779 + 0.626i)8-s + (−0.809 − 0.587i)9-s + (−0.970 + 0.239i)10-s + (−0.970 − 0.239i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (0.779 − 0.626i)15-s + (−0.989 + 0.144i)16-s + (−0.943 − 0.331i)17-s + (0.981 − 0.192i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0307 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0307 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.0307 + 0.999i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (922, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.0307 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3366780601 + 0.3264800806i\)
\(L(\frac12)\) \(\approx\) \(0.3366780601 + 0.3264800806i\)
\(L(1)\) \(\approx\) \(0.6686818039 + 0.01502434031i\)
\(L(1)\) \(\approx\) \(0.6686818039 + 0.01502434031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.681 + 0.732i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.836 + 0.548i)T \)
7 \( 1 + (-0.906 + 0.421i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.943 - 0.331i)T \)
19 \( 1 + (-0.168 - 0.985i)T \)
23 \( 1 + (-0.748 + 0.663i)T \)
29 \( 1 + (0.399 - 0.916i)T \)
31 \( 1 + (-0.443 - 0.896i)T \)
37 \( 1 + (0.399 - 0.916i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (-0.861 + 0.506i)T \)
53 \( 1 + (-0.262 + 0.964i)T \)
59 \( 1 + (-0.168 + 0.985i)T \)
61 \( 1 + (0.485 + 0.873i)T \)
67 \( 1 + (-0.970 - 0.239i)T \)
71 \( 1 + (-0.443 + 0.896i)T \)
73 \( 1 + (-0.998 - 0.0483i)T \)
79 \( 1 + (0.644 + 0.764i)T \)
83 \( 1 + (0.836 + 0.548i)T \)
89 \( 1 + (0.120 + 0.992i)T \)
97 \( 1 + (0.0241 - 0.999i)T \)
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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

−17.51377874069307498032900363739, −16.79164540564695310905712623136, −16.31279047578600823354372421280, −16.11889560181196398909670271159, −14.84698777841234791480749720860, −14.23588334828097619267168941852, −13.48578455170985790977337497064, −12.910790715789839452838329952961, −12.27389120452361999041598886287, −11.48851987747874926066825957308, −10.57315871168736838295872452759, −10.09116470974740579091734764946, −9.76672708457478348932074886728, −9.02148737701218719110605642817, −8.55156283671018714134703847115, −7.80107506160215658378332408429, −6.668442783851953700180036495996, −6.216771931075558999148430308941, −4.86480528255301320179592456443, −4.58962855810497039813974830847, −3.61472939863062823922085505919, −3.05433641295125890849630032428, −2.13761269716701190005931340626, −1.59510958309792862585767503543, −0.184654057875311389840410189930, 0.68367497715345821620189394165, 1.91927646334220191288173675350, 2.4107763021206277944423713467, 2.984694565304081520252720308985, 4.298686136514326085841849477053, 5.50234036392981773758549514679, 5.860339223442925946368659610838, 6.548148777130493318027922240967, 7.12151697078607324457892740207, 7.61114073676985135746563824215, 8.5400945173583433760805779698, 9.27979053602469691365427308213, 9.572638078711727398653795375, 10.37512915787453107962809284785, 11.18597331435737257729323537631, 11.940633170858086148759495065316, 12.90422680106336268657890500282, 13.45266470672487938634666418829, 13.8281271668716444765475886333, 14.792652593801504246898890707340, 15.16798732659905994935093698847, 15.86019094922838762205790927724, 16.764939301036668981492402818610, 17.479047764461235044116638010140, 17.83233361691927727816088140494

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