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Properties

Label 1-6027-6027.2558-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.977 - 0.210i$
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  + (0.599 + 0.800i)2-s + (−0.280 + 0.959i)4-s + (−0.447 − 0.894i)5-s + (−0.936 + 0.351i)8-s + (0.447 − 0.894i)10-s + (−0.420 − 0.907i)11-s + (0.393 − 0.919i)13-s + (−0.842 − 0.538i)16-s + (0.971 + 0.237i)17-s + (0.104 + 0.994i)19-s + (0.983 − 0.178i)20-s + (0.473 − 0.880i)22-s + (0.646 − 0.762i)23-s + (−0.599 + 0.800i)25-s + (0.971 − 0.237i)26-s + ⋯
L(s)  = 1  + (0.599 + 0.800i)2-s + (−0.280 + 0.959i)4-s + (−0.447 − 0.894i)5-s + (−0.936 + 0.351i)8-s + (0.447 − 0.894i)10-s + (−0.420 − 0.907i)11-s + (0.393 − 0.919i)13-s + (−0.842 − 0.538i)16-s + (0.971 + 0.237i)17-s + (0.104 + 0.994i)19-s + (0.983 − 0.178i)20-s + (0.473 − 0.880i)22-s + (0.646 − 0.762i)23-s + (−0.599 + 0.800i)25-s + (0.971 − 0.237i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.977 - 0.210i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (2558, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.977 - 0.210i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.938840539 - 0.2061468124i\)
\(L(\frac12)\) \(\approx\) \(1.938840539 - 0.2061468124i\)
\(L(1)\) \(\approx\) \(1.239845083 + 0.2722713672i\)
\(L(1)\) \(\approx\) \(1.239845083 + 0.2722713672i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.599 + 0.800i)T \)
5 \( 1 + (-0.447 - 0.894i)T \)
11 \( 1 + (-0.420 - 0.907i)T \)
13 \( 1 + (0.393 - 0.919i)T \)
17 \( 1 + (0.971 + 0.237i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (0.646 - 0.762i)T \)
29 \( 1 + (-0.134 + 0.990i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.791 + 0.611i)T \)
43 \( 1 + (-0.963 + 0.266i)T \)
47 \( 1 + (-0.599 - 0.800i)T \)
53 \( 1 + (0.280 - 0.959i)T \)
59 \( 1 + (0.712 + 0.701i)T \)
61 \( 1 + (0.337 - 0.941i)T \)
67 \( 1 + (0.669 - 0.743i)T \)
71 \( 1 + (-0.134 - 0.990i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.992 + 0.119i)T \)
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

−17.80964720786812197361548870068, −17.37607383866786322417058190809, −16.07823061093523712027307655101, −15.67386976824429811550904337244, −14.81901546236042349650218402954, −14.594142347277320631777237137502, −13.60494562941635220743935073063, −13.28792123062957927297973139026, −12.32698161829774113663831609296, −11.64048538829189106464931326722, −11.37330424847559093052248514252, −10.544701560252960444680068179247, −9.870486624249676597179338604655, −9.421816445590280139630798999304, −8.43926854051836397759083382824, −7.45910057685568329036285215241, −6.945013081760213019774590860916, −6.17970624398296163091287293848, −5.38365645138490029122405154741, −4.532275879523269723901343622569, −4.05432653901216208865043153330, −3.12072956966900075256075420407, −2.64277782824005876991061271488, −1.82668014760210182610600294426, −0.86788483157582694616058736112, 0.50026978241647874249113665115, 1.37524371972683629028310392948, 2.79653929496456016259273099155, 3.41211880366996964928016441712, 3.98136740045165027437064563247, 5.03645631448535116760189723427, 5.329465976928317538997895621812, 6.06676306141840231809069726796, 6.81431292993188506179865282789, 7.87749965442991789381655807827, 8.18724683789588754212028173971, 8.58142979666565493263091987340, 9.58780919558131699647469428206, 10.39123283089628197089376633370, 11.29043311184229357952739881564, 11.9933081775049017508778418579, 12.58212014299711103074280354715, 13.17400999107417683191515241374, 13.620434947958659496730488522218, 14.64312719420633306080487032228, 14.96376782821002807418547027425, 15.90033659464744012399347112170, 16.30231196976226438961378909780, 16.77665630841492962090907747200, 17.38893843985936899744307443334

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