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Properties

Label 1-2808-2808.1499-r0-0-0
Degree $1$
Conductor $2808$
Sign $0.560 - 0.828i$
Analytic cond. $13.0402$
Root an. cond. $13.0402$
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  + (−0.173 − 0.984i)5-s + (−0.939 + 0.342i)7-s + (−0.939 + 0.342i)11-s − 17-s + (0.5 + 0.866i)19-s + (−0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (0.766 + 0.642i)29-s + (0.766 − 0.642i)31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + (0.173 + 0.984i)41-s + (0.766 + 0.642i)43-s + (−0.766 − 0.642i)47-s + (0.766 − 0.642i)49-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)5-s + (−0.939 + 0.342i)7-s + (−0.939 + 0.342i)11-s − 17-s + (0.5 + 0.866i)19-s + (−0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (0.766 + 0.642i)29-s + (0.766 − 0.642i)31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + (0.173 + 0.984i)41-s + (0.766 + 0.642i)43-s + (−0.766 − 0.642i)47-s + (0.766 − 0.642i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.560 - 0.828i$
Analytic conductor: \(13.0402\)
Root analytic conductor: \(13.0402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2808,\ (0:\ ),\ 0.560 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7565703435 - 0.4016992421i\)
\(L(\frac12)\) \(\approx\) \(0.7565703435 - 0.4016992421i\)
\(L(1)\) \(\approx\) \(0.7812950358 - 0.09186987765i\)
\(L(1)\) \(\approx\) \(0.7812950358 - 0.09186987765i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 - T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−19.39883758032483094577375476331, −18.68905104321108313804620532156, −17.75969860215498756238802640282, −17.5615496092757819837063836454, −16.20517400714085359852759872067, −15.7424956925204526672653071478, −15.38378473125287681993560382385, −14.168688828214111721519054395187, −13.70969220697812586278855191648, −13.096510208825483719178999490641, −12.1580389489742429439477286311, −11.40028604346882303214488128977, −10.57077667645518731927791250984, −10.22222151033357221231379317886, −9.320362386506466063317588981078, −8.46033467008134509449113261159, −7.51566565185270363612999098927, −6.97485627754046907672225279802, −6.249552359842475585411834852180, −5.507538605453453494770714579426, −4.37476690176007377225573351753, −3.624222176118554062974601488, −2.76367608672652536339730077731, −2.29130621714230581683539766833, −0.65220899981903035361171048087, 0.40639052964076143487577139043, 1.653629074171658949941226225456, 2.552079508998306251061194130498, 3.42971644624766334655696782085, 4.408221064621990271725859824898, 5.01591697033149606416958234313, 5.93391193422680345278241729727, 6.545994601658495039384040109460, 7.64528562773580628384294259027, 8.27199437245266369041322481073, 8.96679443135666461006556050846, 9.84851370192461837447731765223, 10.2278400498752428750302288516, 11.39772975524832360081393565615, 12.169618616284425250020579556300, 12.66812933686995500635248788525, 13.32597185733769946679038730385, 13.94334838093768061570688769045, 15.14735304864847850208189739105, 15.66512342063826596358450069664, 16.27854659991431097013344118806, 16.75998056594960409535215496629, 17.85738007568520690151062006025, 18.26397162720940196366830673043, 19.23118535710192946596493937738

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