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Properties

Label 1-6027-6027.2189-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.670 - 0.741i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (−0.992 + 0.119i)2-s + (0.971 − 0.237i)4-s + (0.998 + 0.0598i)5-s + (−0.936 + 0.351i)8-s + (−0.998 + 0.0598i)10-s + (−0.575 + 0.817i)11-s + (0.393 − 0.919i)13-s + (0.887 − 0.460i)16-s + (−0.280 − 0.959i)17-s + (−0.913 − 0.406i)19-s + (0.983 − 0.178i)20-s + (0.473 − 0.880i)22-s + (0.337 + 0.941i)23-s + (0.992 + 0.119i)25-s + (−0.280 + 0.959i)26-s + ⋯
L(s)  = 1  + (−0.992 + 0.119i)2-s + (0.971 − 0.237i)4-s + (0.998 + 0.0598i)5-s + (−0.936 + 0.351i)8-s + (−0.998 + 0.0598i)10-s + (−0.575 + 0.817i)11-s + (0.393 − 0.919i)13-s + (0.887 − 0.460i)16-s + (−0.280 − 0.959i)17-s + (−0.913 − 0.406i)19-s + (0.983 − 0.178i)20-s + (0.473 − 0.880i)22-s + (0.337 + 0.941i)23-s + (0.992 + 0.119i)25-s + (−0.280 + 0.959i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.670 - 0.741i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (2189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.670 - 0.741i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1781713402 - 0.4014829120i\)
\(L(\frac12)\) \(\approx\) \(0.1781713402 - 0.4014829120i\)
\(L(1)\) \(\approx\) \(0.6958849963 + 0.01277614944i\)
\(L(1)\) \(\approx\) \(0.6958849963 + 0.01277614944i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.992 + 0.119i)T \)
5 \( 1 + (0.998 + 0.0598i)T \)
11 \( 1 + (-0.575 + 0.817i)T \)
13 \( 1 + (0.393 - 0.919i)T \)
17 \( 1 + (-0.280 - 0.959i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (0.337 + 0.941i)T \)
29 \( 1 + (-0.134 + 0.990i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.925 + 0.379i)T \)
43 \( 1 + (-0.963 + 0.266i)T \)
47 \( 1 + (0.992 - 0.119i)T \)
53 \( 1 + (-0.971 + 0.237i)T \)
59 \( 1 + (0.251 - 0.967i)T \)
61 \( 1 + (0.646 + 0.762i)T \)
67 \( 1 + (-0.978 - 0.207i)T \)
71 \( 1 + (-0.134 - 0.990i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.599 + 0.800i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−17.97519912463805586228204700736, −17.17236365423648677661689290411, −16.889163700696001016381129218412, −16.200226577944732995724946652066, −15.54570704543571074593889834472, −14.668130338973180964306629940101, −14.11059591512248511106751209639, −13.193551745703957645751671323839, −12.74318840557330043661477047697, −11.902758155415307816211853253158, −11.033491060244995402774962728535, −10.55627011139401848721388865110, −10.11530325814009687290365527629, −9.138816942539189391055444352086, −8.68655430210140380257148274696, −8.23572819003219728243270560419, −7.18961090526565140007016906319, −6.41679842577238580062310520846, −6.086075181422774165083895047700, −5.250785433450942136716170044311, −4.14555956928257934429989862779, −3.32676544635010034690805390287, −2.353618633189201513089175060069, −1.90973319547363252671242526583, −1.0856032025648073172159976203, 0.14845927669677586109377625322, 1.32389603266797682937262417810, 1.97728611193369302488760638313, 2.703261166178960058882783807197, 3.37285952851818740559350096653, 4.77768923784274720669952954200, 5.39272237529409206703300966802, 6.011214869474501648822881797805, 6.95304136502186821491153385191, 7.259715356253424752820381223420, 8.217373301633453172448972320095, 8.87011264254174588443320899881, 9.54509417230095862038161983805, 10.021301808074815687056728001196, 10.77727829903356760389537174823, 11.14618468644240310921669594124, 12.201437609332932525139064436979, 12.89100936227071191493457278658, 13.44155003064057483264976814146, 14.31097078338977441479205801618, 15.16593375919207076581669883111, 15.44427394414950763130585202990, 16.29842916314854053122693434405, 17.03634595441456683245817424407, 17.530425435559483483833438800039

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