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Properties

Label 1-6027-6027.836-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.919 - 0.393i$
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  + (−0.193 − 0.981i)2-s + (−0.925 + 0.379i)4-s + (−0.772 + 0.635i)5-s + (0.550 + 0.834i)8-s + (0.772 + 0.635i)10-s + (−0.0149 + 0.999i)11-s + (−0.753 + 0.657i)13-s + (0.712 − 0.701i)16-s + (0.791 − 0.611i)17-s + (0.104 + 0.994i)19-s + (0.473 − 0.880i)20-s + (0.983 − 0.178i)22-s + (0.999 + 0.0299i)23-s + (0.193 − 0.981i)25-s + (0.791 + 0.611i)26-s + ⋯
L(s)  = 1  + (−0.193 − 0.981i)2-s + (−0.925 + 0.379i)4-s + (−0.772 + 0.635i)5-s + (0.550 + 0.834i)8-s + (0.772 + 0.635i)10-s + (−0.0149 + 0.999i)11-s + (−0.753 + 0.657i)13-s + (0.712 − 0.701i)16-s + (0.791 − 0.611i)17-s + (0.104 + 0.994i)19-s + (0.473 − 0.880i)20-s + (0.983 − 0.178i)22-s + (0.999 + 0.0299i)23-s + (0.193 − 0.981i)25-s + (0.791 + 0.611i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.919 - 0.393i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (836, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.919 - 0.393i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.255355182 - 0.2572162538i\)
\(L(\frac12)\) \(\approx\) \(1.255355182 - 0.2572162538i\)
\(L(1)\) \(\approx\) \(0.8205116431 - 0.1775398950i\)
\(L(1)\) \(\approx\) \(0.8205116431 - 0.1775398950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.193 - 0.981i)T \)
5 \( 1 + (-0.772 + 0.635i)T \)
11 \( 1 + (-0.0149 + 0.999i)T \)
13 \( 1 + (-0.753 + 0.657i)T \)
17 \( 1 + (0.791 - 0.611i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (0.999 + 0.0299i)T \)
29 \( 1 + (0.691 + 0.722i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.971 - 0.237i)T \)
43 \( 1 + (-0.0448 - 0.998i)T \)
47 \( 1 + (0.193 + 0.981i)T \)
53 \( 1 + (0.925 - 0.379i)T \)
59 \( 1 + (-0.842 + 0.538i)T \)
61 \( 1 + (-0.525 - 0.850i)T \)
67 \( 1 + (0.669 - 0.743i)T \)
71 \( 1 + (0.691 - 0.722i)T \)
73 \( 1 + (0.733 - 0.680i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.946 + 0.323i)T \)
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

−17.507027341898871398035070298103, −16.858920392824796791540054075932, −16.64186182234404797445535524649, −15.58415339549202184391639897949, −15.43173055669078335497003351462, −14.661064258663089983535168016, −13.91638488639415184259254961252, −13.15544470135207135518248683707, −12.729550134145807234303525183977, −11.85391460397282364875667396272, −11.158896491608421226085130831528, −10.28606181438127932258015037267, −9.64131049121876809165638051087, −8.793909616397064921396786838784, −8.32959913555981983470448347468, −7.769577817715556035322510679769, −7.10760105615748257577762575081, −6.27624122613121067454905730591, −5.53349462031401111626274758861, −4.91205313578012581256454442651, −4.31274627538465445667897977421, −3.41391676552889462845552736367, −2.68220758452770729677099616843, −1.01206266719427342781802756258, −0.740334799681621016560705704751, 0.62487344892169160842725593506, 1.61356817792454197497732356843, 2.47661833506305301548699405038, 3.06799937262455679323574372942, 3.820146389890213858709348602838, 4.60028952336329577977239913157, 5.03522564103757630593608627274, 6.235798095649654180474882232353, 7.24237000834893633772521289544, 7.54132009600196815367078379729, 8.347684406977639431889454640317, 9.21302805808605384754036150444, 9.87472722118887845183584378262, 10.32261250706044594527180117518, 11.146004136226892867587344809988, 11.73148928018947015605299034018, 12.38474972036544796486560459330, 12.60525117071937073720460442937, 13.91018219893635182383001453025, 14.21660846501817390491742186322, 14.94917089467792660399690977252, 15.59461928362071932015652183108, 16.66251852739178897716823489028, 16.963862169201280757046264996310, 17.97009514548680227733467442975

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