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Properties

Label 1-33e2-1089.1022-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.288 + 0.957i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (0.0475 + 0.998i)2-s + (−0.995 + 0.0950i)4-s + (−0.928 + 0.371i)5-s + (0.327 + 0.945i)7-s + (−0.142 − 0.989i)8-s + (−0.415 − 0.909i)10-s + (0.995 − 0.0950i)13-s + (−0.928 + 0.371i)14-s + (0.981 − 0.189i)16-s + (−0.959 − 0.281i)17-s + (0.959 − 0.281i)19-s + (0.888 − 0.458i)20-s + (0.327 − 0.945i)23-s + (0.723 − 0.690i)25-s + (0.142 + 0.989i)26-s + ⋯
L(s)  = 1  + (0.0475 + 0.998i)2-s + (−0.995 + 0.0950i)4-s + (−0.928 + 0.371i)5-s + (0.327 + 0.945i)7-s + (−0.142 − 0.989i)8-s + (−0.415 − 0.909i)10-s + (0.995 − 0.0950i)13-s + (−0.928 + 0.371i)14-s + (0.981 − 0.189i)16-s + (−0.959 − 0.281i)17-s + (0.959 − 0.281i)19-s + (0.888 − 0.458i)20-s + (0.327 − 0.945i)23-s + (0.723 − 0.690i)25-s + (0.142 + 0.989i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.288 + 0.957i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.288 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9455035580 + 0.7024892003i\)
\(L(\frac12)\) \(\approx\) \(0.9455035580 + 0.7024892003i\)
\(L(1)\) \(\approx\) \(0.7915401351 + 0.4800104578i\)
\(L(1)\) \(\approx\) \(0.7915401351 + 0.4800104578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0475 + 0.998i)T \)
5 \( 1 + (-0.928 + 0.371i)T \)
7 \( 1 + (0.327 + 0.945i)T \)
13 \( 1 + (0.995 - 0.0950i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (0.235 - 0.971i)T \)
31 \( 1 + (0.580 - 0.814i)T \)
37 \( 1 + (0.415 + 0.909i)T \)
41 \( 1 + (-0.888 - 0.458i)T \)
43 \( 1 + (-0.928 - 0.371i)T \)
47 \( 1 + (0.888 - 0.458i)T \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (-0.0475 + 0.998i)T \)
61 \( 1 + (0.888 - 0.458i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (0.786 + 0.618i)T \)
83 \( 1 + (-0.327 - 0.945i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.928 + 0.371i)T \)
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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

−21.09586763838530864157849720735, −20.30430191947783359616084896050, −19.942869890612703407226901909794, −19.21266713322789373582869399249, −18.23658900574693878741482579649, −17.63686405513538889775116547738, −16.65451527511557986172469651396, −15.870093914113358667302154553691, −14.92246508918345260413826177204, −13.89681788430355331209479257957, −13.38080105546006843870514295433, −12.51331120628552506622244089985, −11.60583169210025842563419515323, −11.07626839734737868742447211762, −10.39684274365787380935257966001, −9.28179856964069392038050232033, −8.52048297132406203739216383300, −7.7817832555129321811943297387, −6.79309959148906668010591512828, −5.3728397243929880779151380229, −4.55074239688815924219395126676, −3.78235390223026763824488890754, −3.165041101468656648698195323833, −1.59472456645292436845250463813, −0.89468320979501479790210421498, 0.71725254573033626011455928482, 2.47666775778179766635390305665, 3.53660894312429407131408965995, 4.45615148426979994627652517551, 5.25962360944310337197337594979, 6.30383071020150016394774046112, 6.91287629251767717924349147252, 8.02391141380746204820081089263, 8.470065454039395971064365293306, 9.272178392874552281094160384098, 10.40923658558533087390868706721, 11.50729778802164940163744383571, 12.01327128926878518458859501959, 13.19236173692767022427644665150, 13.81587100615891782067959359797, 14.91568777852275757920554490560, 15.38668377317791103267626023804, 15.87367280257508857541369030146, 16.76990790692314421988327665742, 17.79532896965224700855303808127, 18.53068085232548036875512460069, 18.81185444831432351404544994494, 19.994101012093898976889714405014, 20.86763209150463904958266172178, 21.93897450977291971833360159226

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