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Properties

Label 1-1080-1080.139-r1-0-0
Degree $1$
Conductor $1080$
Sign $-0.998 - 0.0581i$
Analytic cond. $116.062$
Root an. cond. $116.062$
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)7-s + (−0.939 + 0.342i)11-s + (0.766 − 0.642i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)29-s + (−0.173 − 0.984i)31-s + (−0.5 − 0.866i)37-s + (0.766 − 0.642i)41-s + (0.939 − 0.342i)43-s + (0.173 − 0.984i)47-s + (−0.939 − 0.342i)49-s + 53-s + (−0.939 − 0.342i)59-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)7-s + (−0.939 + 0.342i)11-s + (0.766 − 0.642i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)29-s + (−0.173 − 0.984i)31-s + (−0.5 − 0.866i)37-s + (0.766 − 0.642i)41-s + (0.939 − 0.342i)43-s + (0.173 − 0.984i)47-s + (−0.939 − 0.342i)49-s + 53-s + (−0.939 − 0.342i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.998 - 0.0581i$
Analytic conductor: \(116.062\)
Root analytic conductor: \(116.062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (1:\ ),\ -0.998 - 0.0581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01219956354 - 0.4192718048i\)
\(L(\frac12)\) \(\approx\) \(0.01219956354 - 0.4192718048i\)
\(L(1)\) \(\approx\) \(0.9105459426 - 0.1234681507i\)
\(L(1)\) \(\approx\) \(0.9105459426 - 0.1234681507i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−21.357307906197883151634120036903, −21.15192919375862630668426079773, −20.17154830470113198435863793097, −19.12326753457827956561175886131, −18.4527529089017714311907589333, −18.08280892193247008648039309232, −16.87617834557906730529333181392, −16.05365157635290095262868352713, −15.54597587619247317621809428548, −14.578454337050861692861184520640, −13.82906803477191917396442390437, −12.8979964038663309391351288397, −12.23025520427564620782762166973, −11.20996273391342477493617843682, −10.73084787695663124021294690658, −9.46245392064450546978540424847, −8.82412597818549819890370378342, −8.08141366426975595928595759943, −7.01871371731371411074787337601, −6.11057054872922822887557173098, −5.25205104420790079703856651331, −4.49461900795819533870429081273, −3.11832588515317794316860562100, −2.48223431711947581284763982992, −1.26871233503939908348201670106, 0.0881859228282034515116028428, 1.238452220687984031342314109454, 2.26870365008196202106947709915, 3.63364323594008430742247742433, 4.09147438133771706239482645283, 5.45340452511947591795798170663, 5.989819587597079727130736472697, 7.40079029865744066964477843340, 7.74221205861446829625712782827, 8.713003023772854644296057895721, 9.90713346104936288188410347338, 10.52421804017119129375354549330, 11.11930236035638233956058282905, 12.30881688090370164137681734937, 13.10582547693469474332660824492, 13.641597195377023459225168973479, 14.68540531148548191819060153079, 15.36547152818394753570971659266, 16.233380480501264372160655925267, 17.084018055937532382813103017530, 17.66774237227127738016005105426, 18.586372165232759463839383063004, 19.311929970746757727375171357573, 20.252227491189218119349957462179, 20.889756081924647069051331466565

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