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Properties

Label 1-6017-6017.244-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.474 + 0.880i$
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  + (−0.977 − 0.212i)2-s + (0.995 + 0.0896i)3-s + (0.909 + 0.415i)4-s + (0.817 + 0.576i)5-s + (−0.954 − 0.299i)6-s + (−0.840 + 0.542i)7-s + (−0.800 − 0.598i)8-s + (0.983 + 0.178i)9-s + (−0.675 − 0.736i)10-s + (0.868 + 0.495i)12-s + (0.0448 + 0.998i)13-s + (0.936 − 0.351i)14-s + (0.762 + 0.647i)15-s + (0.655 + 0.755i)16-s + (−0.938 + 0.344i)17-s + (−0.923 − 0.383i)18-s + ⋯
L(s)  = 1  + (−0.977 − 0.212i)2-s + (0.995 + 0.0896i)3-s + (0.909 + 0.415i)4-s + (0.817 + 0.576i)5-s + (−0.954 − 0.299i)6-s + (−0.840 + 0.542i)7-s + (−0.800 − 0.598i)8-s + (0.983 + 0.178i)9-s + (−0.675 − 0.736i)10-s + (0.868 + 0.495i)12-s + (0.0448 + 0.998i)13-s + (0.936 − 0.351i)14-s + (0.762 + 0.647i)15-s + (0.655 + 0.755i)16-s + (−0.938 + 0.344i)17-s + (−0.923 − 0.383i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.474 + 0.880i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.474 + 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8859260319 + 1.484161521i\)
\(L(\frac12)\) \(\approx\) \(0.8859260319 + 1.484161521i\)
\(L(1)\) \(\approx\) \(1.007056656 + 0.3544455158i\)
\(L(1)\) \(\approx\) \(1.007056656 + 0.3544455158i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.977 - 0.212i)T \)
3 \( 1 + (0.995 + 0.0896i)T \)
5 \( 1 + (0.817 + 0.576i)T \)
7 \( 1 + (-0.840 + 0.542i)T \)
13 \( 1 + (0.0448 + 0.998i)T \)
17 \( 1 + (-0.938 + 0.344i)T \)
19 \( 1 + (0.660 + 0.750i)T \)
23 \( 1 + (0.418 + 0.908i)T \)
29 \( 1 + (-0.106 - 0.994i)T \)
31 \( 1 + (0.720 + 0.693i)T \)
37 \( 1 + (-0.843 - 0.536i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.928 + 0.370i)T \)
47 \( 1 + (-0.0724 + 0.997i)T \)
53 \( 1 + (-0.997 - 0.0758i)T \)
59 \( 1 + (0.998 + 0.0483i)T \)
61 \( 1 + (0.828 - 0.559i)T \)
67 \( 1 + (-0.868 - 0.495i)T \)
71 \( 1 + (0.991 + 0.130i)T \)
73 \( 1 + (0.235 - 0.971i)T \)
79 \( 1 + (0.969 + 0.246i)T \)
83 \( 1 + (0.485 + 0.873i)T \)
89 \( 1 + (0.999 - 0.0345i)T \)
97 \( 1 + (-0.628 + 0.777i)T \)
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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

−17.68467340850110207791904567648, −16.79187463486899794520270549706, −16.28135587705781616065683945099, −15.59694464396808416672365332729, −15.086847426510141957049025494736, −14.21488842377221520926494549882, −13.50568518541977492200631230769, −13.05356625590674713692671942437, −12.42400989757614396697856266357, −11.373796021394193127032656942499, −10.42717926809393846611864486263, −10.030134453410332627939952259762, −9.466782499166281339730558952780, −8.76111237914036060603361443464, −8.40965893581151812319666788339, −7.46906483826348086490310638480, −6.80183228114600740283391020939, −6.37894308709481373920594828566, −5.300382029624909495712570543889, −4.60959412372294894132698883511, −3.37359141804429568899315375795, −2.80107155256447398239495769820, −2.14115926187287726871262696771, −1.16620445869673499731424403892, −0.519733080322245711234305939008, 1.25055806764221242409352979064, 2.07335669390172489106880793375, 2.38590680695089675906486537153, 3.37434558467972441962600017269, 3.70625000778564779585852879629, 5.04763283618605997474750494479, 6.150393889080971355301179436421, 6.59452907887209786734397615989, 7.228417644187865820360188594463, 8.02887730853663504780807378918, 8.834965880608694761137802318749, 9.37397492577256588387728513719, 9.687748792446836606366526351815, 10.391613012735053900286117698116, 11.10856114009198292698686318010, 11.95794445269039903861158816730, 12.66179769767401808681774744904, 13.43905280509681167224161031659, 13.93399229841716667391875556990, 14.76699480140442329026280524130, 15.459445150112206881909083053075, 15.9210817913292244705030165491, 16.60048092843010837014390399976, 17.52208168708190822269110188630, 17.94357203211708138221032353146

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