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Properties

Label 1-6027-6027.3134-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.890 + 0.454i$
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  + (0.999 − 0.0299i)2-s + (0.998 − 0.0598i)4-s + (0.0149 − 0.999i)5-s + (0.995 − 0.0896i)8-s + (−0.0149 − 0.999i)10-s + (−0.971 + 0.237i)11-s + (−0.473 + 0.880i)13-s + (0.992 − 0.119i)16-s + (−0.447 − 0.894i)17-s + (0.104 − 0.994i)19-s + (−0.0448 − 0.998i)20-s + (−0.963 + 0.266i)22-s + (−0.887 + 0.460i)23-s + (−0.999 − 0.0299i)25-s + (−0.447 + 0.894i)26-s + ⋯
L(s)  = 1  + (0.999 − 0.0299i)2-s + (0.998 − 0.0598i)4-s + (0.0149 − 0.999i)5-s + (0.995 − 0.0896i)8-s + (−0.0149 − 0.999i)10-s + (−0.971 + 0.237i)11-s + (−0.473 + 0.880i)13-s + (0.992 − 0.119i)16-s + (−0.447 − 0.894i)17-s + (0.104 − 0.994i)19-s + (−0.0448 − 0.998i)20-s + (−0.963 + 0.266i)22-s + (−0.887 + 0.460i)23-s + (−0.999 − 0.0299i)25-s + (−0.447 + 0.894i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.890 + 0.454i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (3134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.890 + 0.454i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06341067034 - 0.2636646813i\)
\(L(\frac12)\) \(\approx\) \(-0.06341067034 - 0.2636646813i\)
\(L(1)\) \(\approx\) \(1.432212150 - 0.3009859797i\)
\(L(1)\) \(\approx\) \(1.432212150 - 0.3009859797i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.999 - 0.0299i)T \)
5 \( 1 + (0.0149 - 0.999i)T \)
11 \( 1 + (-0.971 + 0.237i)T \)
13 \( 1 + (-0.473 + 0.880i)T \)
17 \( 1 + (-0.447 - 0.894i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (-0.887 + 0.460i)T \)
29 \( 1 + (-0.936 + 0.351i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.772 - 0.635i)T \)
43 \( 1 + (0.753 + 0.657i)T \)
47 \( 1 + (-0.999 + 0.0299i)T \)
53 \( 1 + (-0.998 + 0.0598i)T \)
59 \( 1 + (-0.946 + 0.323i)T \)
61 \( 1 + (0.842 - 0.538i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (-0.936 - 0.351i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.525 - 0.850i)T \)
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

−18.048050675707739384145827718, −17.40566089254675177554578635343, −16.672511610638699039165853137738, −15.77017232998965467612166215482, −15.38624524856812130503207668502, −14.81403028372007328345800584528, −14.13795059999468306533478437471, −13.5983693183304646416486666700, −12.81363928587470316384978058528, −12.35796940478259449628673834356, −11.484910148903212344213939958760, −10.89514965104670518431300534162, −10.1621987363450558196422882823, −9.999347068363390239450724060826, −8.3693518482985069655340195764, −7.89105328356495100080169439875, −7.30249657443831647300566207650, −6.37700860916546021384085092620, −5.92793027623708757092176400418, −5.28629156249073178083311558079, −4.3570702476756754324078180027, −3.63180316149323266982326772654, −2.980597320571684315284096117836, −2.33625409364625204712869024285, −1.61709568303637483293222573129, 0.03881011533862319786556131649, 1.33642764605284104319996888245, 2.126117989615418329201390348831, 2.74217754659809246806887768757, 3.71546211167052410302819742628, 4.666912277440565089849354747711, 4.80343092536959388108187985068, 5.59420407696063806569993689533, 6.37684542804005536417707668520, 7.272382629824709976582595746363, 7.67272764940598550408077054146, 8.64498269999969072030822432757, 9.41796620478791511178839802088, 10.01488041074105130734882691532, 10.967194160206916388618495373758, 11.630260852964810789768795469907, 12.07399186864544138242587121769, 12.9703317624047519517945045572, 13.21683190659176368885854261085, 14.00218958480121508035818604477, 14.53333154149214723611626068560, 15.62837953971284392176023337169, 15.84791072939377127554159288043, 16.36551041554311309428438713850, 17.313626216354181025154153853439

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