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Properties

Label 1-33e2-1089.196-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.391 - 0.920i$
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  + (0.380 − 0.924i)2-s + (−0.710 − 0.703i)4-s + (0.820 + 0.572i)5-s + (−0.398 − 0.917i)7-s + (−0.921 + 0.389i)8-s + (0.841 − 0.540i)10-s + (0.988 + 0.151i)13-s + (−0.999 + 0.0190i)14-s + (0.00951 + 0.999i)16-s + (0.696 + 0.717i)17-s + (0.897 + 0.441i)19-s + (−0.179 − 0.983i)20-s + (−0.995 − 0.0950i)23-s + (0.345 + 0.938i)25-s + (0.516 − 0.856i)26-s + ⋯
L(s)  = 1  + (0.380 − 0.924i)2-s + (−0.710 − 0.703i)4-s + (0.820 + 0.572i)5-s + (−0.398 − 0.917i)7-s + (−0.921 + 0.389i)8-s + (0.841 − 0.540i)10-s + (0.988 + 0.151i)13-s + (−0.999 + 0.0190i)14-s + (0.00951 + 0.999i)16-s + (0.696 + 0.717i)17-s + (0.897 + 0.441i)19-s + (−0.179 − 0.983i)20-s + (−0.995 − 0.0950i)23-s + (0.345 + 0.938i)25-s + (0.516 − 0.856i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.391 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.684298389 - 1.114288789i\)
\(L(\frac12)\) \(\approx\) \(1.684298389 - 1.114288789i\)
\(L(1)\) \(\approx\) \(1.256748380 - 0.6225454934i\)
\(L(1)\) \(\approx\) \(1.256748380 - 0.6225454934i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.380 - 0.924i)T \)
5 \( 1 + (0.820 + 0.572i)T \)
7 \( 1 + (-0.398 - 0.917i)T \)
13 \( 1 + (0.988 + 0.151i)T \)
17 \( 1 + (0.696 + 0.717i)T \)
19 \( 1 + (0.897 + 0.441i)T \)
23 \( 1 + (-0.995 - 0.0950i)T \)
29 \( 1 + (0.640 + 0.768i)T \)
31 \( 1 + (0.548 - 0.836i)T \)
37 \( 1 + (0.774 + 0.633i)T \)
41 \( 1 + (-0.432 - 0.901i)T \)
43 \( 1 + (-0.327 + 0.945i)T \)
47 \( 1 + (0.879 - 0.475i)T \)
53 \( 1 + (-0.870 + 0.491i)T \)
59 \( 1 + (0.997 - 0.0760i)T \)
61 \( 1 + (-0.991 - 0.132i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (-0.466 - 0.884i)T \)
73 \( 1 + (-0.736 + 0.676i)T \)
79 \( 1 + (-0.683 - 0.730i)T \)
83 \( 1 + (0.749 - 0.662i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.820 - 0.572i)T \)
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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

−21.73421324667096692917763375377, −21.02434413757382851569907781418, −20.27190177274301305859363523624, −19.00098569094568022849116203359, −18.099211877091891443415106673136, −17.8141878354384883120650840320, −16.71054238358475531741144433418, −15.9512975118611850049609006469, −15.65726218081700293816824678230, −14.41401862354831892679346715356, −13.77147999282974964265933375392, −13.14634261728749402394584879829, −12.273102017685965503927319438554, −11.65101468446326497660495929165, −10.0526426145554249079667103815, −9.425046349035509803697102861, −8.66077681099513584643509987502, −7.95689080025243904404564459354, −6.76118249817747062163901813825, −5.94028445793227260966903762904, −5.47196770070647492983247024644, −4.57331224243185572142299399630, −3.37026943602646464953217911030, −2.48347222311608262466306779867, −0.97510571029318584203334246527, 1.043882000466508351660738511437, 1.83736072756644408344732154923, 3.09694305459202536267110391878, 3.63598821042727906233362077413, 4.650862283538559629569516973881, 5.91468971536757423485491539641, 6.26080571359194126921368671314, 7.54133438452493367269952241916, 8.64554914192644102258631932406, 9.76893517964789775179035408389, 10.15580614506199131735663713804, 10.85974749922009003755658954112, 11.73630119597384148831387356780, 12.709679560154922650857710660133, 13.52157422391952612013422720529, 13.973646654508768595759291870683, 14.642282412100916884619283398222, 15.76906997721393157859985047300, 16.76846089290822374570651454498, 17.60309654223601260391207351271, 18.42473326566986724264003445417, 18.91036159474892598969516459223, 19.96079558508129890116275656481, 20.534733920255009649069694628818, 21.26636051678737592718406181629

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