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Properties

Label 1-6027-6027.2546-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.317 - 0.948i$
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  + (−0.791 + 0.611i)2-s + (0.251 − 0.967i)4-s + (−0.946 − 0.323i)5-s + (0.393 + 0.919i)8-s + (0.946 − 0.323i)10-s + (0.525 + 0.850i)11-s + (0.134 + 0.990i)13-s + (−0.873 − 0.486i)16-s + (−0.712 + 0.701i)17-s + (0.669 − 0.743i)19-s + (−0.550 + 0.834i)20-s + (−0.936 − 0.351i)22-s + (0.447 − 0.894i)23-s + (0.791 + 0.611i)25-s + (−0.712 − 0.701i)26-s + ⋯
L(s)  = 1  + (−0.791 + 0.611i)2-s + (0.251 − 0.967i)4-s + (−0.946 − 0.323i)5-s + (0.393 + 0.919i)8-s + (0.946 − 0.323i)10-s + (0.525 + 0.850i)11-s + (0.134 + 0.990i)13-s + (−0.873 − 0.486i)16-s + (−0.712 + 0.701i)17-s + (0.669 − 0.743i)19-s + (−0.550 + 0.834i)20-s + (−0.936 − 0.351i)22-s + (0.447 − 0.894i)23-s + (0.791 + 0.611i)25-s + (−0.712 − 0.701i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.317 - 0.948i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (2546, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.317 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07112388152 - 0.09883883036i\)
\(L(\frac12)\) \(\approx\) \(0.07112388152 - 0.09883883036i\)
\(L(1)\) \(\approx\) \(0.5417112581 + 0.1469615295i\)
\(L(1)\) \(\approx\) \(0.5417112581 + 0.1469615295i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.791 + 0.611i)T \)
5 \( 1 + (-0.946 - 0.323i)T \)
11 \( 1 + (0.525 + 0.850i)T \)
13 \( 1 + (0.134 + 0.990i)T \)
17 \( 1 + (-0.712 + 0.701i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.447 - 0.894i)T \)
29 \( 1 + (-0.0448 - 0.998i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.842 + 0.538i)T \)
43 \( 1 + (-0.995 - 0.0896i)T \)
47 \( 1 + (-0.791 + 0.611i)T \)
53 \( 1 + (0.251 - 0.967i)T \)
59 \( 1 + (0.575 - 0.817i)T \)
61 \( 1 + (-0.998 - 0.0598i)T \)
67 \( 1 + (-0.913 + 0.406i)T \)
71 \( 1 + (-0.0448 + 0.998i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.925 - 0.379i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−18.101822682577454160230197958846, −17.40616798103990551558798794817, −16.43192461398482801462026137522, −16.25661653616327478740171342445, −15.361700431705274750815147784737, −14.869810731078979179712057554778, −13.721052178240367651983310524462, −13.32626409166544172090488002624, −12.29558785637877085148706571307, −11.87522686376163067300849887127, −11.215269788445337021153812284000, −10.7507966282081227070919297179, −10.03944422508857415685025720583, −9.136994288207840969209558985989, −8.678065397843453333544614210811, −7.86830846833893646151138166536, −7.421671656281183038902067583780, −6.7079068987777428766071426997, −5.788019622741026687599861977127, −4.82676197059738393234543349409, −3.86014878917635685437781177674, −3.296942658569471322925001896852, −2.8796259177751483630841386619, −1.67575421347422639922120635439, −0.86560346887586101602349962113, 0.052284097650972732977794287289, 1.23180494468289138466822332874, 1.85284197805643462036750702939, 2.89888752510085345339674281123, 4.03184200269887851792487521973, 4.61767847055553759896519846611, 5.17875017833512604330772500714, 6.41755664331677645276572669780, 6.81813997984469773729743341591, 7.34620086581513037687225789048, 8.33765545998250241553815418595, 8.65892036668798799943872621460, 9.379542396065683710391314447535, 10.04708916375803979771259655587, 10.89471497596602032210622202910, 11.537062391551581871828912242701, 11.99675366774493543567774839727, 12.92181453496986618512549257413, 13.699970466902102056684227065520, 14.59520006416418302898721788640, 14.9653737528295042269460356501, 15.78093672785208630462102818952, 16.071765361807854964137814741519, 16.98763695230338768927401278312, 17.32460151552484870250560254875

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