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Properties

Label 1-6027-6027.1466-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.682 - 0.731i$
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  + (0.772 − 0.635i)2-s + (0.193 − 0.981i)4-s + (−0.337 + 0.941i)5-s + (−0.473 − 0.880i)8-s + (0.337 + 0.941i)10-s + (0.712 + 0.701i)11-s + (0.936 + 0.351i)13-s + (−0.925 − 0.379i)16-s + (0.946 + 0.323i)17-s + (0.669 + 0.743i)19-s + (0.858 + 0.512i)20-s + (0.995 + 0.0896i)22-s + (−0.0149 − 0.999i)23-s + (−0.772 − 0.635i)25-s + (0.946 − 0.323i)26-s + ⋯
L(s)  = 1  + (0.772 − 0.635i)2-s + (0.193 − 0.981i)4-s + (−0.337 + 0.941i)5-s + (−0.473 − 0.880i)8-s + (0.337 + 0.941i)10-s + (0.712 + 0.701i)11-s + (0.936 + 0.351i)13-s + (−0.925 − 0.379i)16-s + (0.946 + 0.323i)17-s + (0.669 + 0.743i)19-s + (0.858 + 0.512i)20-s + (0.995 + 0.0896i)22-s + (−0.0149 − 0.999i)23-s + (−0.772 − 0.635i)25-s + (0.946 − 0.323i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.682 - 0.731i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1466, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.682 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.915542154 - 1.267331799i\)
\(L(\frac12)\) \(\approx\) \(2.915542154 - 1.267331799i\)
\(L(1)\) \(\approx\) \(1.658345033 - 0.4694938176i\)
\(L(1)\) \(\approx\) \(1.658345033 - 0.4694938176i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.772 - 0.635i)T \)
5 \( 1 + (-0.337 + 0.941i)T \)
11 \( 1 + (0.712 + 0.701i)T \)
13 \( 1 + (0.936 + 0.351i)T \)
17 \( 1 + (0.946 + 0.323i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.0149 - 0.999i)T \)
29 \( 1 + (-0.393 - 0.919i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.992 + 0.119i)T \)
43 \( 1 + (-0.691 - 0.722i)T \)
47 \( 1 + (0.772 - 0.635i)T \)
53 \( 1 + (0.193 - 0.981i)T \)
59 \( 1 + (-0.280 + 0.959i)T \)
61 \( 1 + (0.873 - 0.486i)T \)
67 \( 1 + (-0.913 - 0.406i)T \)
71 \( 1 + (-0.393 + 0.919i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.163 - 0.986i)T \)
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

−17.68161311370364083209290752793, −16.82603144934145625729985339766, −16.349055309941090113662937089539, −15.9418613420509712517064067649, −15.25597717350121887238122859674, −14.4936042348044936766477468530, −13.73764388274702111488751204520, −13.38653745905281425234538252104, −12.61197387807419866534355742999, −11.98021449093767879552861543038, −11.43733860773290656905147950087, −10.80397947399274476195992142761, −9.466989960477984751458596358897, −9.02553109038668354146013449943, −8.31596749494053130251249711365, −7.69635532437257448357419889844, −7.044645507753163412072338101947, −6.08804594377710009729909511150, −5.57484497912378160000207107058, −4.95627509630041408181556115670, −4.17408823916300233571608196920, −3.38716114117452149371360208770, −3.04233951362386732150981156559, −1.54000151527035692391497409809, −0.89642903759714276888608881113, 0.76687154038922129730501582495, 1.70869936200013580374075693980, 2.38689372386624469144035543671, 3.25891643535897339707132306979, 3.95485771247871643195079698586, 4.24287016941606486897306994949, 5.452101288564835648861826559935, 6.08880624611628797564165511153, 6.63507861746638447660232797919, 7.401413687590775522671929238464, 8.17947125660522320390989572243, 9.19364275405429009997079688716, 10.012862647219870398594175475668, 10.28979047994231955716893703717, 11.23587696445321333257618205085, 11.72429501261964169166731367844, 12.169969771140509423031496597846, 13.05202294058802183725390077185, 13.74123932494482237886677527079, 14.366455423785996541895121386209, 14.86791896427032479675342916936, 15.333683640714024104102692863770, 16.23133288178122512583540949998, 16.825933512699692443693343050130, 17.88353095824703031202301544139

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