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Properties

Label 1-6017-6017.1630-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.999 + 0.00691i$
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  + (−0.136 − 0.990i)2-s + (−0.309 + 0.951i)3-s + (−0.962 + 0.270i)4-s + (0.581 − 0.813i)5-s + (0.984 + 0.176i)6-s + (−0.827 + 0.561i)7-s + (0.399 + 0.916i)8-s + (−0.809 − 0.587i)9-s + (−0.885 − 0.464i)10-s + (0.0402 − 0.999i)12-s + (0.104 − 0.994i)13-s + (0.669 + 0.743i)14-s + (0.594 + 0.804i)15-s + (0.853 − 0.520i)16-s + (0.0563 − 0.998i)17-s + (−0.471 + 0.881i)18-s + ⋯
L(s)  = 1  + (−0.136 − 0.990i)2-s + (−0.309 + 0.951i)3-s + (−0.962 + 0.270i)4-s + (0.581 − 0.813i)5-s + (0.984 + 0.176i)6-s + (−0.827 + 0.561i)7-s + (0.399 + 0.916i)8-s + (−0.809 − 0.587i)9-s + (−0.885 − 0.464i)10-s + (0.0402 − 0.999i)12-s + (0.104 − 0.994i)13-s + (0.669 + 0.743i)14-s + (0.594 + 0.804i)15-s + (0.853 − 0.520i)16-s + (0.0563 − 0.998i)17-s + (−0.471 + 0.881i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.999 + 0.00691i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1630, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.999 + 0.00691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002687272864 - 0.7768766157i\)
\(L(\frac12)\) \(\approx\) \(0.002687272864 - 0.7768766157i\)
\(L(1)\) \(\approx\) \(0.6757357316 - 0.3491652134i\)
\(L(1)\) \(\approx\) \(0.6757357316 - 0.3491652134i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.136 - 0.990i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.581 - 0.813i)T \)
7 \( 1 + (-0.827 + 0.561i)T \)
13 \( 1 + (0.104 - 0.994i)T \)
17 \( 1 + (0.0563 - 0.998i)T \)
19 \( 1 + (0.231 - 0.972i)T \)
23 \( 1 + (0.919 + 0.391i)T \)
29 \( 1 + (-0.981 + 0.192i)T \)
31 \( 1 + (0.943 - 0.331i)T \)
37 \( 1 + (0.657 + 0.753i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (-0.845 - 0.534i)T \)
47 \( 1 + (0.818 - 0.574i)T \)
53 \( 1 + (0.737 - 0.675i)T \)
59 \( 1 + (0.231 + 0.972i)T \)
61 \( 1 + (-0.984 - 0.176i)T \)
67 \( 1 + (-0.845 - 0.534i)T \)
71 \( 1 + (-0.184 - 0.982i)T \)
73 \( 1 + (0.00805 - 0.999i)T \)
79 \( 1 + (-0.989 - 0.144i)T \)
83 \( 1 + (-0.414 - 0.910i)T \)
89 \( 1 + (0.970 + 0.239i)T \)
97 \( 1 + (-0.704 - 0.709i)T \)
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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

−18.02317257466973905459669396066, −17.13376986605347302605490306339, −16.93974894828402524287098451002, −16.34142410506009622355138006355, −15.38813229773306581237145924118, −14.61126734958183834236790158653, −14.136543538278582570058672158809, −13.53881505069675575053250087696, −13.01121509188199054287131309811, −12.39706945126741138155582773287, −11.39301634308325417014211322835, −10.60633820596178018872887281614, −10.052841234032629348736042362045, −9.30996480190028563429612689210, −8.546048991625184894262700857381, −7.74490084267272892172224822792, −7.00940951118407628705584751164, −6.75095314842956487826349403736, −5.95991673247666615405177502522, −5.69053523859499298725257369587, −4.47771614468321540026666794247, −3.68445660218147881752402538739, −2.84526428562988856565330955571, −1.76452275441711632714559094214, −1.09262883162517159629273263111, 0.27464579341547979817752374509, 0.94956281777021567731153139831, 2.17263808225609774927405442814, 3.03158176045412334175123761783, 3.31610599638088348860752463241, 4.47546246780607599323745138963, 5.04545598081204667240547073603, 5.49970263998918473512750312943, 6.262710422460585588980162974428, 7.42877809348493728297241374968, 8.53813399668443327755279256724, 8.99182555322661058160594245191, 9.470546579383866225172599993713, 10.08229885608966762869317529811, 10.56332071462744572985626092289, 11.67722994671961992983623004005, 11.798447568703496332105393645107, 12.76872438412141659748425302695, 13.39094278966556597319616458227, 13.694034044517456862439937787619, 15.03186474236677310457036644497, 15.32208324962480187694351548861, 16.2637762555142323647120174697, 16.82134081063879918009852364405, 17.328396564336914284266175663445

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