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Properties

Label 1-6027-6027.1007-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.280 - 0.959i$
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  + (0.0448 − 0.998i)2-s + (−0.995 − 0.0896i)4-s + (−0.691 + 0.722i)5-s + (−0.134 + 0.990i)8-s + (0.691 + 0.722i)10-s + (0.936 + 0.351i)11-s + (−0.0448 + 0.998i)13-s + (0.983 + 0.178i)16-s + (0.995 − 0.0896i)17-s + (−0.809 − 0.587i)19-s + (0.753 − 0.657i)20-s + (0.393 − 0.919i)22-s + (−0.753 − 0.657i)23-s + (−0.0448 − 0.998i)25-s + (0.995 + 0.0896i)26-s + ⋯
L(s)  = 1  + (0.0448 − 0.998i)2-s + (−0.995 − 0.0896i)4-s + (−0.691 + 0.722i)5-s + (−0.134 + 0.990i)8-s + (0.691 + 0.722i)10-s + (0.936 + 0.351i)11-s + (−0.0448 + 0.998i)13-s + (0.983 + 0.178i)16-s + (0.995 − 0.0896i)17-s + (−0.809 − 0.587i)19-s + (0.753 − 0.657i)20-s + (0.393 − 0.919i)22-s + (−0.753 − 0.657i)23-s + (−0.0448 − 0.998i)25-s + (0.995 + 0.0896i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.280 - 0.959i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.280 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6756421108 - 0.9012528470i\)
\(L(\frac12)\) \(\approx\) \(0.6756421108 - 0.9012528470i\)
\(L(1)\) \(\approx\) \(0.8122802464 - 0.3318018964i\)
\(L(1)\) \(\approx\) \(0.8122802464 - 0.3318018964i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.0448 - 0.998i)T \)
5 \( 1 + (-0.691 + 0.722i)T \)
11 \( 1 + (0.936 + 0.351i)T \)
13 \( 1 + (-0.0448 + 0.998i)T \)
17 \( 1 + (0.995 - 0.0896i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.753 - 0.657i)T \)
29 \( 1 + (0.858 + 0.512i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.858 + 0.512i)T \)
43 \( 1 + (0.473 - 0.880i)T \)
47 \( 1 + (0.0448 - 0.998i)T \)
53 \( 1 + (-0.995 - 0.0896i)T \)
59 \( 1 + (0.473 - 0.880i)T \)
61 \( 1 + (-0.753 + 0.657i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (-0.623 - 0.781i)T \)
79 \( 1 - T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.0448 + 0.998i)T \)
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

−17.51367839032827569252823093028, −17.287351995523911766282920409595, −16.37830069200292515222647236644, −16.091129262058660049336820862907, −15.38348127995465659431630073193, −14.62436704675916945869062024842, −14.23279155876421481343970316444, −13.37183833051713038134088939391, −12.529520933168017114929139987, −12.34438920368220318861185753724, −11.424275822294364461549743921409, −10.483002251596291478736204356604, −9.70778415279872131934299451299, −9.09045487492288437416973489890, −8.29710249049808600374332702341, −7.92054615545721129388531798969, −7.33257799031127216783676816579, −6.22043907624843018980253175864, −5.88916548286877585018251645941, −5.03077973906110104284594006013, −4.30361079944074421293276139276, −3.712353974729323972946499173517, −3.0212123057531658255694445708, −1.43884017817152560514221687051, −0.819126702103139452473525569465, 0.38615418744948753150000044405, 1.46305433296412062671766874376, 2.26407393977401120101513435974, 2.93576303068417568169848140245, 3.89124188644767194066431689635, 4.1737575402111683754685901894, 4.94965712001990908334315412936, 6.0954971877462547034252204819, 6.68331344911653876298070084652, 7.50777189392305271586726623794, 8.3052099268587791742378557954, 8.94510706750718487418215023903, 9.715465502119207903111823707427, 10.27602536875224289007224012025, 11.032513974422166911879637545397, 11.606930946248973163301985025584, 12.10747245050981119186518524204, 12.62861117175191209546197108344, 13.649801803248464321072803738389, 14.253115053262446064788096049447, 14.68286683202563880825445184677, 15.31718471226256053458560047972, 16.40561781085097087025671353670, 16.87884031073034602961326755763, 17.72576193199480113355704445151

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