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Properties

Label 1-6027-6027.5648-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.188 - 0.982i$
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  + (0.691 + 0.722i)2-s + (−0.0448 + 0.998i)4-s + (−0.393 − 0.919i)5-s + (−0.753 + 0.657i)8-s + (0.393 − 0.919i)10-s + (0.983 + 0.178i)11-s + (−0.691 − 0.722i)13-s + (−0.995 − 0.0896i)16-s + (0.0448 + 0.998i)17-s + (0.309 − 0.951i)19-s + (0.936 − 0.351i)20-s + (0.550 + 0.834i)22-s + (−0.936 − 0.351i)23-s + (−0.691 + 0.722i)25-s + (0.0448 − 0.998i)26-s + ⋯
L(s)  = 1  + (0.691 + 0.722i)2-s + (−0.0448 + 0.998i)4-s + (−0.393 − 0.919i)5-s + (−0.753 + 0.657i)8-s + (0.393 − 0.919i)10-s + (0.983 + 0.178i)11-s + (−0.691 − 0.722i)13-s + (−0.995 − 0.0896i)16-s + (0.0448 + 0.998i)17-s + (0.309 − 0.951i)19-s + (0.936 − 0.351i)20-s + (0.550 + 0.834i)22-s + (−0.936 − 0.351i)23-s + (−0.691 + 0.722i)25-s + (0.0448 − 0.998i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.188 - 0.982i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (5648, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.188 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4119455865 - 0.4985314898i\)
\(L(\frac12)\) \(\approx\) \(0.4119455865 - 0.4985314898i\)
\(L(1)\) \(\approx\) \(1.115319885 + 0.2921040016i\)
\(L(1)\) \(\approx\) \(1.115319885 + 0.2921040016i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.691 + 0.722i)T \)
5 \( 1 + (-0.393 - 0.919i)T \)
11 \( 1 + (0.983 + 0.178i)T \)
13 \( 1 + (-0.691 - 0.722i)T \)
17 \( 1 + (0.0448 + 0.998i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.936 - 0.351i)T \)
29 \( 1 + (-0.963 - 0.266i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.963 - 0.266i)T \)
43 \( 1 + (0.858 - 0.512i)T \)
47 \( 1 + (0.691 + 0.722i)T \)
53 \( 1 + (-0.0448 + 0.998i)T \)
59 \( 1 + (0.858 - 0.512i)T \)
61 \( 1 + (-0.936 + 0.351i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (-0.963 + 0.266i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.691 - 0.722i)T \)
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

−18.17308232169674728824052821315, −17.314365936847093933862607532104, −16.43194906310498867780772409396, −15.797842462169587570684185373686, −15.033233626335481586504885853920, −14.38977463731950343800007288271, −14.084339628718680883756948650377, −13.46814461462019953740462204484, −12.36686057684687198449214798317, −11.81283626272608125486680994988, −11.57492363233299577046212803024, −10.77718002618693070754673533501, −9.89737602065860883249542101750, −9.62949658219211048286713482138, −8.71406826413123083830359124981, −7.62717123967012582844070320252, −7.01506354786227534500646531559, −6.345003733390171426497655069056, −5.67060505526237476395090441051, −4.80344075004501080843985622714, −3.886229061358723743473348204, −3.66674036426159050739606760727, −2.65647320072093729230284717647, −2.06241933539342962300402614022, −1.15846206142882130247116760954, 0.12577548482486417472109173644, 1.332759755992426528858410956676, 2.33564849111611366436694533724, 3.273186348584741256029701204717, 4.117112132188398457082160266077, 4.435125002706450074080424282671, 5.36941981890203331403750002527, 5.84487439498778120983959127491, 6.73315510297817504304897368819, 7.43334226865193634906421123059, 8.02949481205571200743600090678, 8.75537963874433361225906913835, 9.24221520167711692625007883429, 10.16152390955304112855548193699, 11.167399354346802829388376728541, 11.88103240643687742897688474487, 12.50539827725278368712654777368, 12.73906398725664727420847914757, 13.74017501655680323230870304457, 14.19371588382415842363151691470, 15.100217004324282378535859197131, 15.46035088128701506594282115439, 16.12273491084083830733697201581, 16.87230369010682380870973839739, 17.419728373204176859918622964033

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