Properties

Label 1-33e2-1089.164-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.999 + 0.00576i$
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  + (−0.786 + 0.618i)2-s + (0.235 − 0.971i)4-s + (−0.580 − 0.814i)5-s + (−0.0475 + 0.998i)7-s + (0.415 + 0.909i)8-s + (0.959 + 0.281i)10-s + (−0.235 + 0.971i)13-s + (−0.580 − 0.814i)14-s + (−0.888 − 0.458i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (−0.928 + 0.371i)20-s + (−0.0475 − 0.998i)23-s + (−0.327 + 0.945i)25-s + (−0.415 − 0.909i)26-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (0.235 − 0.971i)4-s + (−0.580 − 0.814i)5-s + (−0.0475 + 0.998i)7-s + (0.415 + 0.909i)8-s + (0.959 + 0.281i)10-s + (−0.235 + 0.971i)13-s + (−0.580 − 0.814i)14-s + (−0.888 − 0.458i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (−0.928 + 0.371i)20-s + (−0.0475 − 0.998i)23-s + (−0.327 + 0.945i)25-s + (−0.415 − 0.909i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.00576i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.999 + 0.00576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7539004827 + 0.002174889612i\)
\(L(\frac12)\) \(\approx\) \(0.7539004827 + 0.002174889612i\)
\(L(1)\) \(\approx\) \(0.6490935096 + 0.09363325691i\)
\(L(1)\) \(\approx\) \(0.6490935096 + 0.09363325691i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.786 + 0.618i)T \)
5 \( 1 + (-0.580 - 0.814i)T \)
7 \( 1 + (-0.0475 + 0.998i)T \)
13 \( 1 + (-0.235 + 0.971i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (0.981 + 0.189i)T \)
31 \( 1 + (0.723 - 0.690i)T \)
37 \( 1 + (-0.959 - 0.281i)T \)
41 \( 1 + (0.928 + 0.371i)T \)
43 \( 1 + (-0.580 + 0.814i)T \)
47 \( 1 + (-0.928 + 0.371i)T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (0.786 + 0.618i)T \)
61 \( 1 + (-0.928 + 0.371i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (-0.841 + 0.540i)T \)
79 \( 1 + (0.995 - 0.0950i)T \)
83 \( 1 + (0.0475 - 0.998i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.580 - 0.814i)T \)
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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

−21.30027965502275509581321537424, −20.4183705821858158728759805650, −19.69290223928216975198734669760, −19.35912928207180783487242075992, −18.33934911109327385989750445401, −17.59238331817649415903187028900, −17.128717287573152841467259957758, −15.91581049930870360004896203863, −15.503051889553464143614156239858, −14.2830811829812645207134988124, −13.51529980730175156129401313579, −12.5424258305363834628453455780, −11.77802438475594810055795924259, −10.90024105150511835205402749750, −10.35148414595023580521903618450, −9.778854277028067013165763550448, −8.42547119820896677414244994564, −7.82836545655352162749938338809, −7.12104226218595390582035294716, −6.30040114942913371141680806421, −4.74937883077123396569001746952, −3.59703019690043181367635242126, −3.269442201015397332746584263759, −1.98691701323568125597626813377, −0.79605575293329713490887027123, 0.59569240924526805168951192805, 1.86341866142403355313786606007, 2.85262888532170898097209825774, 4.60528591599302553536847127991, 4.93444835276303531206959366306, 6.146074093178964490017342046777, 6.89385935173626032027923142475, 7.87051081291614344780862667625, 8.70581812855824942348818322177, 9.15536220244146939101828761343, 9.929253943635576510857312117321, 11.32171092886157151854718798607, 11.68288314307527620285780787063, 12.67201847960940296174683536320, 13.74777640284562086016344468284, 14.61226909711810928048138653707, 15.51624489396546430243310348726, 16.02975390606777874678299812117, 16.582299374253123465482151998351, 17.6321189084588483408277413556, 18.2211140589420551664800128303, 19.173275553474825516428038200104, 19.5890049190861183547858889894, 20.513202485327435222276461341, 21.257558992537592647342808717970

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