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Properties

Label 1-6017-6017.3625-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: $\chi_{6017} (3625, \cdot )$
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

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

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