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Properties

Label 1-6017-6017.692-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.833 + 0.551i$
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  + (−0.845 + 0.534i)2-s − 3-s + (0.428 − 0.903i)4-s + (0.919 − 0.391i)5-s + (0.845 − 0.534i)6-s + (−0.0402 + 0.999i)7-s + (0.120 + 0.992i)8-s + 9-s + (−0.568 + 0.822i)10-s + (−0.428 + 0.903i)12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.919 + 0.391i)15-s + (−0.632 − 0.774i)16-s + (0.948 − 0.316i)17-s + (−0.845 + 0.534i)18-s + ⋯
L(s)  = 1  + (−0.845 + 0.534i)2-s − 3-s + (0.428 − 0.903i)4-s + (0.919 − 0.391i)5-s + (0.845 − 0.534i)6-s + (−0.0402 + 0.999i)7-s + (0.120 + 0.992i)8-s + 9-s + (−0.568 + 0.822i)10-s + (−0.428 + 0.903i)12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.919 + 0.391i)15-s + (−0.632 − 0.774i)16-s + (0.948 − 0.316i)17-s + (−0.845 + 0.534i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.833 + 0.551i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (692, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.833 + 0.551i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.130405764 + 0.3401296238i\)
\(L(\frac12)\) \(\approx\) \(1.130405764 + 0.3401296238i\)
\(L(1)\) \(\approx\) \(0.7246094270 + 0.1253352374i\)
\(L(1)\) \(\approx\) \(0.7246094270 + 0.1253352374i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.845 + 0.534i)T \)
3 \( 1 - T \)
5 \( 1 + (0.919 - 0.391i)T \)
7 \( 1 + (-0.0402 + 0.999i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.948 - 0.316i)T \)
19 \( 1 + (0.632 - 0.774i)T \)
23 \( 1 + (-0.278 - 0.960i)T \)
29 \( 1 + (0.970 - 0.239i)T \)
31 \( 1 + (0.354 - 0.935i)T \)
37 \( 1 + (-0.278 + 0.960i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.996 + 0.0804i)T \)
47 \( 1 + (0.692 + 0.721i)T \)
53 \( 1 + (0.799 + 0.600i)T \)
59 \( 1 + (0.632 + 0.774i)T \)
61 \( 1 + (-0.845 + 0.534i)T \)
67 \( 1 + (-0.996 + 0.0804i)T \)
71 \( 1 + (-0.987 - 0.160i)T \)
73 \( 1 + (0.919 + 0.391i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (0.799 + 0.600i)T \)
89 \( 1 + (-0.885 + 0.464i)T \)
97 \( 1 + (-0.200 + 0.979i)T \)
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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

−17.81063344765181989585290427377, −17.07428698954100846104007464004, −16.51817669824267979349927153264, −16.198707735581972752904835099990, −15.24038203704558713363208201660, −13.972921906201690621064126695900, −13.81218608906255022687291445891, −12.87552951589205923742515115864, −12.15278592945867997214371853205, −11.61982223187364426896671555651, −10.86080225761547498430445089130, −10.18056667484911710456382935004, −10.1021774298977323552280578329, −9.24398943556780777416021353108, −8.36062402641812137249259492245, −7.34969705170205624491138377219, −7.03021725279069199993221855472, −6.27291852140924237897381409570, −5.59290849999543040955073007162, −4.66583952441713797989361134418, −3.66310909113852852211438910400, −3.2949330691887043540659134102, −1.74205022912217453151943565411, −1.61701061626892218406683266086, −0.649053789972966420172735809216, 0.81892698948550408926624687413, 1.22256865079844727604866987339, 2.32612011749918445562312604485, 2.96962155861640381547352953561, 4.59212780859588045635892403377, 5.14364562437917756520505217837, 5.74993209041407779694473717430, 6.189937813382957702668859686377, 6.805134150618419151072811599711, 7.81184842834113324885213921319, 8.433504139832775224909552931857, 9.16879744479415394655136547055, 9.89265807462552633223653178102, 10.22929299792803841098215207921, 11.014165461678913662347163464131, 11.91104502504197094007686932001, 12.20977906878356393967774651022, 13.27558322170616307974826209027, 13.737032739907206073410086197961, 14.84074557754734243589554186825, 15.337156676816007501551935397812, 16.037139521073868510109748691351, 16.55826701831553782215766596033, 17.1244893849448089058774489235, 17.834193572677191271068835410823

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