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Properties

Label 1-33e2-1089.527-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.941 + 0.336i$
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  + (0.928 + 0.371i)2-s + (0.723 + 0.690i)4-s + (0.995 − 0.0950i)5-s + (0.888 − 0.458i)7-s + (0.415 + 0.909i)8-s + (0.959 + 0.281i)10-s + (−0.723 − 0.690i)13-s + (0.995 − 0.0950i)14-s + (0.0475 + 0.998i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (0.786 + 0.618i)20-s + (0.888 + 0.458i)23-s + (0.981 − 0.189i)25-s + (−0.415 − 0.909i)26-s + ⋯
L(s)  = 1  + (0.928 + 0.371i)2-s + (0.723 + 0.690i)4-s + (0.995 − 0.0950i)5-s + (0.888 − 0.458i)7-s + (0.415 + 0.909i)8-s + (0.959 + 0.281i)10-s + (−0.723 − 0.690i)13-s + (0.995 − 0.0950i)14-s + (0.0475 + 0.998i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (0.786 + 0.618i)20-s + (0.888 + 0.458i)23-s + (0.981 − 0.189i)25-s + (−0.415 − 0.909i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.941 + 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.591670952 + 0.6226303460i\)
\(L(\frac12)\) \(\approx\) \(3.591670952 + 0.6226303460i\)
\(L(1)\) \(\approx\) \(2.272628701 + 0.3671653061i\)
\(L(1)\) \(\approx\) \(2.272628701 + 0.3671653061i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.928 + 0.371i)T \)
5 \( 1 + (0.995 - 0.0950i)T \)
7 \( 1 + (0.888 - 0.458i)T \)
13 \( 1 + (-0.723 - 0.690i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (-0.327 - 0.945i)T \)
31 \( 1 + (0.235 + 0.971i)T \)
37 \( 1 + (-0.959 - 0.281i)T \)
41 \( 1 + (-0.786 + 0.618i)T \)
43 \( 1 + (0.995 + 0.0950i)T \)
47 \( 1 + (0.786 + 0.618i)T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (-0.928 + 0.371i)T \)
61 \( 1 + (0.786 + 0.618i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (-0.841 + 0.540i)T \)
79 \( 1 + (-0.580 - 0.814i)T \)
83 \( 1 + (-0.888 + 0.458i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (-0.995 - 0.0950i)T \)
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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

−21.5094377890479285744015253500, −20.74296277636945034804522491140, −20.24023781796691013004171024303, −18.95353606478450293430772451415, −18.56706917679784563990815833354, −17.37756604642765436652477855280, −16.84714265982902296540776210800, −15.65645529164021847218343201315, −14.83766533077966915976806143894, −14.28740407914535677275880007858, −13.65371989371748875861610977102, −12.67901570604167385913290781129, −12.06752359032699332590206595804, −11.104151358007736100717268012812, −10.483158415676447116622121895584, −9.53041335065803826197174746296, −8.71693790590154835242845071762, −7.398158485258760645983870196355, −6.56385164434020220587464009938, −5.64702629020276192985471710442, −5.04134160414489181304798284727, −4.17456027702083879583961755625, −2.93701424477510432914300888525, −2.0319748590390468120458713106, −1.47164908506782423048558164744, 1.25772860280046002736834157633, 2.37505319797143942886047833985, 3.10748477079480939546362555790, 4.519149992864697805919026025799, 5.05184693137376997504049340700, 5.74619059607182220621619572293, 6.93787447594104848265197812334, 7.414539981413725311063384191161, 8.50889683561139773017422034672, 9.4679122624306564603823111641, 10.5469435652607359782224025072, 11.25148961995366384717446511981, 12.12606153177879618302661278078, 13.10291975178163310237038114598, 13.678652686197715556747674077944, 14.277293543497046816187026963029, 15.108698027428038044017166429130, 15.84845949002168131080296247607, 16.9004862955440328056780941477, 17.57741865752711040327298268311, 17.85983042798647960615034649027, 19.35974077959392286006274236907, 20.37599548448439179341920755381, 20.744873325360181740919129571164, 21.57515830667198230012466842650

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