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Properties

Label 1-1080-1080.947-r1-0-0
Degree $1$
Conductor $1080$
Sign $-0.524 - 0.851i$
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  + (−0.642 − 0.766i)7-s + (−0.173 − 0.984i)11-s + (−0.342 − 0.939i)13-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (0.642 − 0.766i)23-s + (0.939 + 0.342i)29-s + (−0.766 − 0.642i)31-s + (0.866 + 0.5i)37-s + (0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (0.642 + 0.766i)47-s + (−0.173 + 0.984i)49-s − i·53-s + (0.173 − 0.984i)59-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)7-s + (−0.173 − 0.984i)11-s + (−0.342 − 0.939i)13-s + (0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (0.642 − 0.766i)23-s + (0.939 + 0.342i)29-s + (−0.766 − 0.642i)31-s + (0.866 + 0.5i)37-s + (0.939 − 0.342i)41-s + (0.984 − 0.173i)43-s + (0.642 + 0.766i)47-s + (−0.173 + 0.984i)49-s − i·53-s + (0.173 − 0.984i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.524 - 0.851i$
Analytic conductor: \(116.062\)
Root analytic conductor: \(116.062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (947, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (1:\ ),\ -0.524 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7857822453 - 1.406448804i\)
\(L(\frac12)\) \(\approx\) \(0.7857822453 - 1.406448804i\)
\(L(1)\) \(\approx\) \(0.9753608850 - 0.2863128091i\)
\(L(1)\) \(\approx\) \(0.9753608850 - 0.2863128091i\)

Euler product

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

−21.55291040501080705247040148376, −20.8700391578672044201616791147, −19.76794410515203531466019310653, −19.333277029450387953413386934416, −18.3646878648322547140676160575, −17.78565333649821273302755315619, −16.77554904110470397808740700205, −16.01672669193957933157307357561, −15.33945007243675610419851088962, −14.523195592131632672828801136617, −13.6690509956741044446208265178, −12.712640620287273601632749424, −12.10295469292063106182930468218, −11.36961982081919325794860594998, −10.219189098575889992181390700, −9.35584209820740338081402283283, −9.03076997073593347151850731362, −7.56037067337456574473215982881, −7.069557702489127876396363091101, −5.99336531874955146378319712351, −5.12857042114243225839686856771, −4.26645819742223549303661739180, −3.01293482953365594534315616546, −2.337136634965502451250330864485, −1.069597461001982539293916678028, 0.39783042067530926914226842620, 1.16628678986523594795674615176, 2.79272231866987513889937415107, 3.39924147660541090142321182171, 4.39073456493987984753231244796, 5.61750826685530193283923231154, 6.16381384264966156547254157567, 7.36872918737338255742726998243, 7.95219951167970383111388812770, 8.94578022322866864554769715556, 10.02302942752988425279843408212, 10.48613213981165136628890648427, 11.37220967095854570769390128078, 12.582018384624222502410454670381, 12.93450428174366808923461335128, 14.03697714548161298921926150586, 14.549088816942322599470563390247, 15.674280219802003282733460932106, 16.43180324552295718111143193710, 16.93227847264534359163816152755, 17.88713911493008963529412527894, 18.85185399492983064152245866829, 19.35059560921308760880193835388, 20.30542914850667948968866103343, 20.85929785623652353230765377644

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