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Properties

Label 1-6017-6017.145-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.928 - 0.370i$
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  + (0.997 + 0.0643i)2-s + (−0.309 + 0.951i)3-s + (0.991 + 0.128i)4-s + (−0.513 + 0.857i)5-s + (−0.369 + 0.929i)6-s + (−0.962 + 0.270i)7-s + (0.981 + 0.192i)8-s + (−0.809 − 0.587i)9-s + (−0.568 + 0.822i)10-s + (−0.428 + 0.903i)12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (−0.657 − 0.753i)15-s + (0.966 + 0.254i)16-s + (−0.581 + 0.813i)17-s + (−0.769 − 0.638i)18-s + ⋯
L(s)  = 1  + (0.997 + 0.0643i)2-s + (−0.309 + 0.951i)3-s + (0.991 + 0.128i)4-s + (−0.513 + 0.857i)5-s + (−0.369 + 0.929i)6-s + (−0.962 + 0.270i)7-s + (0.981 + 0.192i)8-s + (−0.809 − 0.587i)9-s + (−0.568 + 0.822i)10-s + (−0.428 + 0.903i)12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (−0.657 − 0.753i)15-s + (0.966 + 0.254i)16-s + (−0.581 + 0.813i)17-s + (−0.769 − 0.638i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.928 - 0.370i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.928 - 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6427301373 - 0.1235811979i\)
\(L(\frac12)\) \(\approx\) \(0.6427301373 - 0.1235811979i\)
\(L(1)\) \(\approx\) \(0.9792488130 + 0.5564463378i\)
\(L(1)\) \(\approx\) \(0.9792488130 + 0.5564463378i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.997 + 0.0643i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.513 + 0.857i)T \)
7 \( 1 + (-0.962 + 0.270i)T \)
13 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (-0.581 + 0.813i)T \)
19 \( 1 + (-0.541 - 0.840i)T \)
23 \( 1 + (-0.278 - 0.960i)T \)
29 \( 1 + (0.527 + 0.849i)T \)
31 \( 1 + (-0.836 + 0.548i)T \)
37 \( 1 + (-0.999 + 0.0322i)T \)
41 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.996 + 0.0804i)T \)
47 \( 1 + (0.899 - 0.435i)T \)
53 \( 1 + (-0.293 - 0.955i)T \)
59 \( 1 + (-0.541 + 0.840i)T \)
61 \( 1 + (0.369 - 0.929i)T \)
67 \( 1 + (-0.996 + 0.0804i)T \)
71 \( 1 + (0.892 - 0.450i)T \)
73 \( 1 + (-0.0884 + 0.996i)T \)
79 \( 1 + (0.0241 - 0.999i)T \)
83 \( 1 + (-0.999 - 0.0161i)T \)
89 \( 1 + (-0.885 + 0.464i)T \)
97 \( 1 + (0.737 - 0.675i)T \)
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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

−17.493251523858553336983532230208, −16.96987702377062027840196343241, −16.459662737456705035137121531912, −15.68721376587346991127693412555, −15.2678381174027682786698636240, −14.14226995538024079914489708993, −13.71129895462952452392880253803, −12.99697742597032292682105169379, −12.56784888516469457075490990373, −12.06887731335266571071923024124, −11.52186984871494046376398577946, −10.688272080770270942560420791262, −9.895234067812117049779290786831, −9.062705049734078016969925133319, −8.07138105423364191445948152216, −7.4135468705011382785058765234, −7.05164233433305596476482235533, −6.06903887195958349533105937510, −5.62320283129478904996155697472, −4.84821629581736959060901372693, −4.09556847001090001565800882389, −3.33340053851936709587107109548, −2.49826622926965333515047661652, −1.78061332833321028860481733503, −0.78100889897749004571919359490, 0.13891571097835614864816101607, 2.00335902943633200725250590245, 2.76804217574358369398839794659, 3.27607163265671743226246659003, 4.02605040645090474962744419328, 4.58215494498515942440737384985, 5.3223752157154224068199068211, 6.25093620350830440230953972146, 6.66042916754262913672497383470, 7.186606307099685323112138825040, 8.36549730401732447177448747436, 9.02702242152884395227298511595, 10.07520877134144908892429769572, 10.46600944462534120451583075149, 11.08186165415087148498079149018, 11.76689088273284726861490200496, 12.425581277860400601433891721316, 12.91624682642908771060770320267, 14.00505226627462793758362168940, 14.51075775349923788501336290310, 15.13453753426346934512461622985, 15.51673860564002391994493864049, 16.20319489775035563971164130449, 16.702675994299602881444992442088, 17.439697748600134480102529106781

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