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Properties

Label 1-6017-6017.107-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.485 + 0.874i$
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  + (−0.0724 + 0.997i)2-s + (0.809 − 0.587i)3-s + (−0.989 − 0.144i)4-s + (−0.399 + 0.916i)5-s + (0.527 + 0.849i)6-s + (0.644 + 0.764i)7-s + (0.215 − 0.976i)8-s + (0.309 − 0.951i)9-s + (−0.885 − 0.464i)10-s + (−0.885 + 0.464i)12-s + (−0.309 + 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.215 + 0.976i)15-s + (0.958 + 0.285i)16-s + (0.779 − 0.626i)17-s + (0.926 + 0.377i)18-s + ⋯
L(s)  = 1  + (−0.0724 + 0.997i)2-s + (0.809 − 0.587i)3-s + (−0.989 − 0.144i)4-s + (−0.399 + 0.916i)5-s + (0.527 + 0.849i)6-s + (0.644 + 0.764i)7-s + (0.215 − 0.976i)8-s + (0.309 − 0.951i)9-s + (−0.885 − 0.464i)10-s + (−0.885 + 0.464i)12-s + (−0.309 + 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.215 + 0.976i)15-s + (0.958 + 0.285i)16-s + (0.779 − 0.626i)17-s + (0.926 + 0.377i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.485 + 0.874i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.485 + 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.045011695 + 1.204018885i\)
\(L(\frac12)\) \(\approx\) \(2.045011695 + 1.204018885i\)
\(L(1)\) \(\approx\) \(1.202716326 + 0.5707752882i\)
\(L(1)\) \(\approx\) \(1.202716326 + 0.5707752882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.0724 + 0.997i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.399 + 0.916i)T \)
7 \( 1 + (0.644 + 0.764i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.779 - 0.626i)T \)
19 \( 1 + (0.943 + 0.331i)T \)
23 \( 1 + (-0.120 - 0.992i)T \)
29 \( 1 + (0.681 - 0.732i)T \)
31 \( 1 + (0.607 + 0.794i)T \)
37 \( 1 + (0.681 - 0.732i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.885 - 0.464i)T \)
47 \( 1 + (0.485 + 0.873i)T \)
53 \( 1 + (-0.861 + 0.506i)T \)
59 \( 1 + (0.943 - 0.331i)T \)
61 \( 1 + (-0.527 - 0.849i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (0.607 - 0.794i)T \)
73 \( 1 + (-0.995 + 0.0965i)T \)
79 \( 1 + (-0.168 - 0.985i)T \)
83 \( 1 + (0.399 - 0.916i)T \)
89 \( 1 + (0.970 + 0.239i)T \)
97 \( 1 + (-0.998 + 0.0483i)T \)
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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

−17.50305869085864448631078124633, −17.10301523219240949967960397804, −16.350206728725782426352815399436, −15.559273463970190498021726171672, −14.88728441865549402717300465922, −14.19436647105701475734375392972, −13.52853055771637875813620477364, −13.10176326538741277368999745751, −12.30364779459559396495599681503, −11.596390704841132845275451504481, −10.98232592768362631867112463882, −10.00804048940421862458477765999, −9.94853087818965229081255889001, −8.97816679618542776583269677750, −8.28134220245132816041539264861, −7.89146350607947234281958929766, −7.310448405665720913961696714751, −5.53169429654505814712055580654, −5.1283982962215988499541280913, −4.41747196877048455854403019579, −3.765223598930638950642241445892, −3.24241279572407298735767191919, −2.352034658090102745162849078174, −1.32451587250511624044783488643, −0.89534929526379722583978626622, 0.73144134949214658973127091323, 1.77495326712995135993386262301, 2.67098142729857911797672025698, 3.28563797338105230831358605929, 4.24628993816578872199206162617, 4.872200097206138358848911849785, 5.93649812833591971392507186939, 6.46614651946566189900654423764, 7.17904375998956136381939982487, 7.78615340988352490456143615285, 8.16866801290800533966217981944, 9.027375190847097579198412705590, 9.56765524338252130620815612335, 10.2895607491490675555773132828, 11.38245123818965632928293764231, 12.12727119152622140981006229476, 12.462567131883938254780753927919, 13.725986938029370178968765243587, 14.1267373997076881686706170613, 14.471692889864043774735631147087, 15.0653751915831193215744377270, 15.82147220439869392319358656071, 16.22549194708926886650116699136, 17.37912950343593518662688054869, 17.878253125687881570572906440633

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