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Properties

Label 1-6017-6017.398-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.986 - 0.162i$
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  + (0.986 − 0.164i)2-s + (−0.858 − 0.512i)3-s + (0.945 − 0.325i)4-s + (0.181 − 0.983i)5-s + (−0.931 − 0.364i)6-s + (−0.405 − 0.914i)7-s + (0.878 − 0.476i)8-s + (0.473 + 0.880i)9-s + (0.0172 − 0.999i)10-s + (−0.978 − 0.205i)12-s + (0.963 − 0.266i)13-s + (−0.550 − 0.834i)14-s + (−0.660 + 0.750i)15-s + (0.788 − 0.615i)16-s + (0.411 − 0.911i)17-s + (0.612 + 0.790i)18-s + ⋯
L(s)  = 1  + (0.986 − 0.164i)2-s + (−0.858 − 0.512i)3-s + (0.945 − 0.325i)4-s + (0.181 − 0.983i)5-s + (−0.931 − 0.364i)6-s + (−0.405 − 0.914i)7-s + (0.878 − 0.476i)8-s + (0.473 + 0.880i)9-s + (0.0172 − 0.999i)10-s + (−0.978 − 0.205i)12-s + (0.963 − 0.266i)13-s + (−0.550 − 0.834i)14-s + (−0.660 + 0.750i)15-s + (0.788 − 0.615i)16-s + (0.411 − 0.911i)17-s + (0.612 + 0.790i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.986 - 0.162i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.986 - 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2159221572 - 2.645076420i\)
\(L(\frac12)\) \(\approx\) \(0.2159221572 - 2.645076420i\)
\(L(1)\) \(\approx\) \(1.236997376 - 0.9995848969i\)
\(L(1)\) \(\approx\) \(1.236997376 - 0.9995848969i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.986 - 0.164i)T \)
3 \( 1 + (-0.858 - 0.512i)T \)
5 \( 1 + (0.181 - 0.983i)T \)
7 \( 1 + (-0.405 - 0.914i)T \)
13 \( 1 + (0.963 - 0.266i)T \)
17 \( 1 + (0.411 - 0.911i)T \)
19 \( 1 + (0.0999 + 0.994i)T \)
23 \( 1 + (0.985 - 0.171i)T \)
29 \( 1 + (-0.996 - 0.0827i)T \)
31 \( 1 + (-0.361 + 0.932i)T \)
37 \( 1 + (-0.556 - 0.830i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.449 + 0.893i)T \)
47 \( 1 + (0.399 + 0.916i)T \)
53 \( 1 + (0.0930 - 0.995i)T \)
59 \( 1 + (-0.715 - 0.698i)T \)
61 \( 1 + (0.295 + 0.955i)T \)
67 \( 1 + (0.978 + 0.205i)T \)
71 \( 1 + (-0.954 - 0.299i)T \)
73 \( 1 + (-0.975 - 0.219i)T \)
79 \( 1 + (0.161 - 0.986i)T \)
83 \( 1 + (-0.262 - 0.964i)T \)
89 \( 1 + (-0.322 - 0.946i)T \)
97 \( 1 + (-0.328 + 0.944i)T \)
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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

−17.98956690866709263231723062939, −17.06246669780045374577343894361, −16.74650774623007603551912828033, −15.73808688843192405577112527986, −15.31901326786971104209424014215, −15.03832395474517917373782363559, −14.188137758166592187040032566644, −13.274399588295864621460289166396, −12.88860301401881371373891366587, −11.998734073773012975333979733161, −11.41247218532145929517402017962, −10.97530415029929493324212138015, −10.3673171083969661586226372179, −9.489609510745480033345360619915, −8.784176314526876477871848711358, −7.69318250024831218453933610252, −6.79875767377042942306304048999, −6.474246246219266631475565377000, −5.63282318436869409942732821641, −5.455945147370789220916898819143, −4.31992267320366368844217266265, −3.6482452613468304664140877038, −3.07918151211405439056051371186, −2.24478960425509393428994281583, −1.29290247449834351005435830726, 0.526629390662032920166623525708, 1.26010561028813520828016517747, 1.79549544823464274799013110251, 3.03894627160780419806059505968, 3.80552489136620215495528439495, 4.4932013634466806181092772078, 5.22407207943455531693702946394, 5.7674104433725881797842096455, 6.33163522119286095433340128447, 7.336387453868026016274685482211, 7.52745400585903092500532120824, 8.67419000250977485397414303653, 9.63975680435549896690565581107, 10.37320065241254736144352164485, 10.95940060369064315475194817030, 11.55789172419789636369801442729, 12.40092705149704860934288840364, 12.78270109771611483008514618091, 13.291588549592656982070291838710, 13.91663268520707706138961840192, 14.48527700393100335098271008727, 15.74002849411136871192009270555, 16.214020395443916846010488619838, 16.48796828693953769191526024206, 17.25909263440353669078918235207

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