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Properties

Label 1-6017-6017.1724-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.492 - 0.870i$
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  + (−0.769 + 0.638i)2-s + (0.809 − 0.587i)3-s + (0.184 − 0.982i)4-s + (−0.0884 + 0.996i)5-s + (−0.247 + 0.968i)6-s + (−0.554 + 0.832i)7-s + (0.485 + 0.873i)8-s + (0.309 − 0.951i)9-s + (−0.568 − 0.822i)10-s + (−0.428 − 0.903i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (0.513 + 0.857i)15-s + (−0.932 − 0.362i)16-s + (−0.00805 + 0.999i)17-s + (0.369 + 0.929i)18-s + ⋯
L(s)  = 1  + (−0.769 + 0.638i)2-s + (0.809 − 0.587i)3-s + (0.184 − 0.982i)4-s + (−0.0884 + 0.996i)5-s + (−0.247 + 0.968i)6-s + (−0.554 + 0.832i)7-s + (0.485 + 0.873i)8-s + (0.309 − 0.951i)9-s + (−0.568 − 0.822i)10-s + (−0.428 − 0.903i)12-s + (0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (0.513 + 0.857i)15-s + (−0.932 − 0.362i)16-s + (−0.00805 + 0.999i)17-s + (0.369 + 0.929i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.492 - 0.870i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.492 - 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006872294198 + 0.01178606088i\)
\(L(\frac12)\) \(\approx\) \(0.006872294198 + 0.01178606088i\)
\(L(1)\) \(\approx\) \(0.7255345301 + 0.2331280017i\)
\(L(1)\) \(\approx\) \(0.7255345301 + 0.2331280017i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.769 + 0.638i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.0884 + 0.996i)T \)
7 \( 1 + (-0.554 + 0.832i)T \)
13 \( 1 + (0.978 - 0.207i)T \)
17 \( 1 + (-0.00805 + 0.999i)T \)
19 \( 1 + (-0.966 - 0.254i)T \)
23 \( 1 + (-0.278 + 0.960i)T \)
29 \( 1 + (-0.644 - 0.764i)T \)
31 \( 1 + (0.998 - 0.0483i)T \)
37 \( 1 + (-0.339 + 0.940i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.996 - 0.0804i)T \)
47 \( 1 + (-0.136 + 0.990i)T \)
53 \( 1 + (-0.324 - 0.945i)T \)
59 \( 1 + (-0.966 + 0.254i)T \)
61 \( 1 + (0.247 - 0.968i)T \)
67 \( 1 + (-0.996 - 0.0804i)T \)
71 \( 1 + (-0.457 - 0.889i)T \)
73 \( 1 + (-0.974 - 0.223i)T \)
79 \( 1 + (-0.607 - 0.794i)T \)
83 \( 1 + (0.818 + 0.574i)T \)
89 \( 1 + (-0.885 - 0.464i)T \)
97 \( 1 + (-0.993 - 0.112i)T \)
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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

−18.01861184344178931422710897008, −17.098450874809762558835959798304, −16.5686756285865533906684676570, −16.1269972701163847700099383471, −15.70408110243387025352088642264, −14.62408917045413380131328333235, −13.698523831055688355906225189424, −13.37520605165833491335890291458, −12.676340896398554291368020998496, −12.0189184318642295954050309130, −11.014235169389710059252855736964, −10.57751918697209488350772363115, −9.860885568767358646617134177630, −9.26874681057347380815056058924, −8.670778466680680317352731776168, −8.260051810790599007915467267143, −7.416681806673036998260331334489, −6.769868484956605751215460879501, −5.6720571019620372841558792854, −4.38486894707059635567763678469, −4.283965944011830144945319680834, −3.43843821452997638249759607569, −2.71146271271231512152207194539, −1.791648571890988809014152317920, −1.05274748922781571952885503663, 0.004004726182339623734350985374, 1.454591482210173261359316410567, 2.002911537892897129675194724817, 2.86436755285763926880300109459, 3.476133043669515188764836986248, 4.45070730660335891511541885945, 5.89532500701409313335065828237, 6.1602166299660613236826351813, 6.66393273580657326239327996929, 7.53657493852678216706999673225, 8.19778713558801429413807111445, 8.56755955444276102832325053588, 9.46630630835259175505099030877, 9.91890140166691895340152690050, 10.751702942590237854563510343841, 11.43254425324252489175562182855, 12.19299705490701506134546785976, 13.25397042289114512617487538543, 13.52258519268071933232114519179, 14.48868830283509676031664188981, 15.04396501022792441862205919072, 15.4200176747864245611487681527, 15.923177604527813568374829813291, 17.00813860667012704073608665572, 17.72898737564671770869428881511

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