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Properties

Label 1-1080-1080.1067-r1-0-0
Degree $1$
Conductor $1080$
Sign $-0.996 - 0.0880i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.996 - 0.0880i$
Analytic conductor: \(116.062\)
Root analytic conductor: \(116.062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (1:\ ),\ -0.996 - 0.0880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02265447118 - 0.5136465203i\)
\(L(\frac12)\) \(\approx\) \(0.02265447118 - 0.5136465203i\)
\(L(1)\) \(\approx\) \(0.9511895929 - 0.1388635246i\)
\(L(1)\) \(\approx\) \(0.9511895929 - 0.1388635246i\)

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

−21.57641755567452965463452555990, −21.04560531524150270000854726926, −20.109237641289082818201461203978, −19.224302935619071912363906089437, −18.42572995983736839943623397402, −18.001584880773633039016831250135, −16.923569771640740746435159535261, −16.04354033250570417737035074276, −15.59125077442887527903784224826, −14.37519547819623395812673528859, −13.97120221418809905881614789558, −13.030293176751110753303371818055, −11.886091695187165868881604587421, −11.49858243056789173547644328225, −10.63687359097546117193549234548, −9.48759867672490239466022202297, −8.76818221054500661813988901809, −8.08969855166523260080654927204, −7.060800943505150495032053102244, −6.01439178542469447105583230833, −5.379257035220091145597093718414, −4.32573030781536031193601425818, −3.35280205546856309787441222652, −2.23991677209166625843547201443, −1.37238365223830828120961782874, 0.10278376706740803110074299872, 1.27478645214660187660226343190, 2.28149927974733367241947718127, 3.426656608554296587909506156032, 4.50859589800853413694716870024, 5.02874514982155531674720197385, 6.316580890124566051813809684330, 7.11886375739656583782662558110, 7.96178525665575721830346665451, 8.679962299196029323375872466346, 9.85429771472070845879655413348, 10.53913954350384156020414472534, 11.207861813893031174294497085188, 12.248424632909419536053348159859, 13.03647200327464692797353422395, 13.773958115686382576925704355160, 14.653499408391623820333014716169, 15.39245264571033250261594900170, 16.11814797715037846098627810869, 17.224720502037783458931803500406, 17.806271915040381009756993050021, 18.23226953821286304793746494314, 19.710399308579640277844038859919, 20.023552682260346558714852864033, 20.77225420632572733332787419131

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