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Properties

Label 1-6027-6027.3407-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.416 + 0.908i$
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  + (0.447 − 0.894i)2-s + (−0.599 − 0.800i)4-s + (0.525 − 0.850i)5-s + (−0.983 + 0.178i)8-s + (−0.525 − 0.850i)10-s + (−0.842 − 0.538i)11-s + (−0.550 − 0.834i)13-s + (−0.280 + 0.959i)16-s + (−0.992 − 0.119i)17-s + (0.669 − 0.743i)19-s + (−0.995 + 0.0896i)20-s + (−0.858 + 0.512i)22-s + (−0.420 − 0.907i)23-s + (−0.447 − 0.894i)25-s + (−0.992 + 0.119i)26-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)2-s + (−0.599 − 0.800i)4-s + (0.525 − 0.850i)5-s + (−0.983 + 0.178i)8-s + (−0.525 − 0.850i)10-s + (−0.842 − 0.538i)11-s + (−0.550 − 0.834i)13-s + (−0.280 + 0.959i)16-s + (−0.992 − 0.119i)17-s + (0.669 − 0.743i)19-s + (−0.995 + 0.0896i)20-s + (−0.858 + 0.512i)22-s + (−0.420 − 0.907i)23-s + (−0.447 − 0.894i)25-s + (−0.992 + 0.119i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.416 + 0.908i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (3407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.416 + 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6649669636 - 0.4265302397i\)
\(L(\frac12)\) \(\approx\) \(-0.6649669636 - 0.4265302397i\)
\(L(1)\) \(\approx\) \(0.6189376081 - 0.8189040131i\)
\(L(1)\) \(\approx\) \(0.6189376081 - 0.8189040131i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.447 - 0.894i)T \)
5 \( 1 + (0.525 - 0.850i)T \)
11 \( 1 + (-0.842 - 0.538i)T \)
13 \( 1 + (-0.550 - 0.834i)T \)
17 \( 1 + (-0.992 - 0.119i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.420 - 0.907i)T \)
29 \( 1 + (0.753 - 0.657i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.946 - 0.323i)T \)
43 \( 1 + (0.134 + 0.990i)T \)
47 \( 1 + (0.447 - 0.894i)T \)
53 \( 1 + (-0.599 - 0.800i)T \)
59 \( 1 + (-0.925 - 0.379i)T \)
61 \( 1 + (-0.575 - 0.817i)T \)
67 \( 1 + (-0.913 + 0.406i)T \)
71 \( 1 + (0.753 + 0.657i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.998 - 0.0598i)T \)
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

−18.07834711035466864771573089381, −17.55004344302055717238816630161, −16.97392641784893464789229390040, −16.1680380693339019084175858117, −15.42184879755462387831235624098, −15.14694448702817892086245121437, −14.20701481864468750185986093898, −13.81636250020695391606771525961, −13.32171464655494773625599036625, −12.33516906684550757617957444678, −11.9161404552544943210725753964, −10.92238253465664698911812174203, −10.230654255388376211968464937787, −9.46521762617990800988576314287, −8.98866223095920033232360345723, −7.82364019293521199877173064750, −7.49037628230909301403017265409, −6.75369166261982954856320128110, −6.1591928083771308327023415, −5.441705140602788385773319466, −4.75195050876026858697790655827, −4.00235447889594942259837795604, −3.14893600765507248303320037893, −2.43658131102846229408852140054, −1.65537935920248955048517351729, 0.186506204917659551721287767477, 0.86245341075253291998348039602, 1.90828738185945196437825119866, 2.60217118219732519311091002984, 3.15549168579800347173721146267, 4.2611410553036285751652443520, 4.93771787585511185277904881876, 5.29155263738998754663194788237, 6.10173318845186287035128971166, 6.8862526749315970740625221884, 8.16021144380759293047875210835, 8.50045220611796551629875204304, 9.38624341714988557748395846834, 9.91192241489338299593865554204, 10.64355744630738599775345048953, 11.13599242846902480272387624411, 12.14018553171469385727728546855, 12.53631940075801279180130101124, 13.168444533691934101727380570525, 13.795295317531500272997150918090, 14.15503621172390566068536722964, 15.34281447243147370812279201933, 15.66681226293349847639538003508, 16.48394329808281032648045958122, 17.45151612026460515153248230020

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