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Properties

Label 1-33e2-1089.452-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.974 + 0.224i$
Analytic cond. $117.029$
Root an. cond. $117.029$
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  + (−0.235 − 0.971i)2-s + (−0.888 + 0.458i)4-s + (0.327 + 0.945i)5-s + (−0.995 + 0.0950i)7-s + (0.654 + 0.755i)8-s + (0.841 − 0.540i)10-s + (−0.888 + 0.458i)13-s + (0.327 + 0.945i)14-s + (0.580 − 0.814i)16-s + (0.142 + 0.989i)17-s + (−0.142 + 0.989i)19-s + (−0.723 − 0.690i)20-s + (0.995 + 0.0950i)23-s + (−0.786 + 0.618i)25-s + (0.654 + 0.755i)26-s + ⋯
L(s)  = 1  + (−0.235 − 0.971i)2-s + (−0.888 + 0.458i)4-s + (0.327 + 0.945i)5-s + (−0.995 + 0.0950i)7-s + (0.654 + 0.755i)8-s + (0.841 − 0.540i)10-s + (−0.888 + 0.458i)13-s + (0.327 + 0.945i)14-s + (0.580 − 0.814i)16-s + (0.142 + 0.989i)17-s + (−0.142 + 0.989i)19-s + (−0.723 − 0.690i)20-s + (0.995 + 0.0950i)23-s + (−0.786 + 0.618i)25-s + (0.654 + 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.974 + 0.224i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (452, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.974 + 0.224i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06864446585 + 0.6036459171i\)
\(L(\frac12)\) \(\approx\) \(0.06864446585 + 0.6036459171i\)
\(L(1)\) \(\approx\) \(0.7114299701 + 0.02723620091i\)
\(L(1)\) \(\approx\) \(0.7114299701 + 0.02723620091i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.235 - 0.971i)T \)
5 \( 1 + (0.327 + 0.945i)T \)
7 \( 1 + (-0.995 + 0.0950i)T \)
13 \( 1 + (-0.888 + 0.458i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (-0.928 + 0.371i)T \)
31 \( 1 + (0.0475 + 0.998i)T \)
37 \( 1 + (0.841 - 0.540i)T \)
41 \( 1 + (-0.723 + 0.690i)T \)
43 \( 1 + (-0.327 + 0.945i)T \)
47 \( 1 + (-0.723 - 0.690i)T \)
53 \( 1 + (-0.415 + 0.909i)T \)
59 \( 1 + (-0.235 + 0.971i)T \)
61 \( 1 + (0.723 + 0.690i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (0.415 + 0.909i)T \)
79 \( 1 + (0.981 + 0.189i)T \)
83 \( 1 + (0.995 - 0.0950i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (-0.327 + 0.945i)T \)
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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

−20.74723675136008376824230310977, −20.0312079901178439276610884729, −19.20675373970442545124575022345, −18.52447503959695605572542708967, −17.35433965821135106235441716189, −17.05627219487593441801678176575, −16.25344303933302110587855030519, −15.563395658672384776189412688980, −14.82677999073533629545622276159, −13.70735389535396150084722062583, −13.132732352605934294868389451801, −12.58021903906957086493961519520, −11.356730613967178160599312454655, −10.00398668323353076161808203850, −9.54302983817084533989902091057, −8.92247723081780726608892997698, −7.88787604137339864081516967406, −7.07628592463940020116649289252, −6.285785093487875323396940989248, −5.24662784215697477967009471895, −4.80536632870622447368926173470, −3.58781312145563617493524554975, −2.31884833862265700551897820145, −0.73723622763695765052364648354, −0.19206183210394272075961985055, 1.41329396536467161211838074649, 2.36917785299627720201902854053, 3.21546042461208567831253186675, 3.86397578560712579641705749553, 5.13143336687840192699445935031, 6.18277200939061071254046020072, 7.054905412722513603856873317887, 7.99718808558993802389640850490, 9.14949462995617651494252224130, 9.77383221858128134879857047627, 10.42770646060222365221241841475, 11.150596132685006902253553629668, 12.1544237678157984775869167393, 12.8049482973058713906411389833, 13.55686033898865648757669713989, 14.54151825734749406187894697894, 15.04219544949401569105586220921, 16.53770577477671985685872994709, 16.9490625117960708831806780389, 18.02226283845587258733667866651, 18.63860263045348014074054621907, 19.3839970461896700114650354810, 19.730651882886536955834285845523, 20.94892341751003404637785684120, 21.69315371853790568232609483980

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