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Properties

Label 1-33e2-1089.778-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.291 + 0.956i$
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  + (−0.123 + 0.992i)2-s + (−0.969 − 0.244i)4-s + (0.964 − 0.263i)5-s + (0.761 − 0.647i)7-s + (0.362 − 0.931i)8-s + (0.142 + 0.989i)10-s + (0.0665 + 0.997i)13-s + (0.548 + 0.836i)14-s + (0.879 + 0.475i)16-s + (0.736 − 0.676i)17-s + (−0.198 − 0.980i)19-s + (−0.999 + 0.0190i)20-s + (0.235 + 0.971i)23-s + (0.861 − 0.508i)25-s + (−0.998 − 0.0570i)26-s + ⋯
L(s)  = 1  + (−0.123 + 0.992i)2-s + (−0.969 − 0.244i)4-s + (0.964 − 0.263i)5-s + (0.761 − 0.647i)7-s + (0.362 − 0.931i)8-s + (0.142 + 0.989i)10-s + (0.0665 + 0.997i)13-s + (0.548 + 0.836i)14-s + (0.879 + 0.475i)16-s + (0.736 − 0.676i)17-s + (−0.198 − 0.980i)19-s + (−0.999 + 0.0190i)20-s + (0.235 + 0.971i)23-s + (0.861 − 0.508i)25-s + (−0.998 − 0.0570i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.291 + 0.956i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (778, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.291 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.136266150 + 1.582900694i\)
\(L(\frac12)\) \(\approx\) \(2.136266150 + 1.582900694i\)
\(L(1)\) \(\approx\) \(1.181175709 + 0.5066404434i\)
\(L(1)\) \(\approx\) \(1.181175709 + 0.5066404434i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.123 + 0.992i)T \)
5 \( 1 + (0.964 - 0.263i)T \)
7 \( 1 + (0.761 - 0.647i)T \)
13 \( 1 + (0.0665 + 0.997i)T \)
17 \( 1 + (0.736 - 0.676i)T \)
19 \( 1 + (-0.198 - 0.980i)T \)
23 \( 1 + (0.235 + 0.971i)T \)
29 \( 1 + (-0.00951 + 0.999i)T \)
31 \( 1 + (0.345 + 0.938i)T \)
37 \( 1 + (-0.466 + 0.884i)T \)
41 \( 1 + (0.290 + 0.956i)T \)
43 \( 1 + (-0.0475 - 0.998i)T \)
47 \( 1 + (0.820 + 0.572i)T \)
53 \( 1 + (-0.0285 + 0.999i)T \)
59 \( 1 + (-0.683 - 0.730i)T \)
61 \( 1 + (-0.797 + 0.603i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (0.993 + 0.113i)T \)
73 \( 1 + (-0.610 - 0.791i)T \)
79 \( 1 + (-0.161 - 0.986i)T \)
83 \( 1 + (-0.997 + 0.0760i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.964 + 0.263i)T \)
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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

−21.07369465745071204583721420187, −20.60943672369280201377183509314, −19.51164119945021978002900108645, −18.54884941082105566902140295184, −18.30877813881132317202302526962, −17.28079540947579199041303043843, −16.92861921141899282925059454058, −15.39059964885535920428822733626, −14.53420122132359501559611985955, −14.050650392261215206433695493161, −12.93694983065152485689398961820, −12.47156161951910987109515668298, −11.52581672164398802948307913954, −10.567883682387588590024162791677, −10.16877429876945243662693157052, −9.2031173255854017545412255864, −8.35460568278044106884335361664, −7.689458242108447087465784676861, −6.01193781615504790458774650406, −5.5659814316221397315255453920, −4.52702729399571518069630024562, −3.41836645043854177209687751033, −2.4236949597307355525337322823, −1.80210251996575280564451682250, −0.70353129119250397964263714117, 0.93123896165003990807237804444, 1.64773213487595329639244204727, 3.19371975753561609029317968320, 4.56270913569839719759314444506, 4.974968584563070758339787386610, 5.939876576252099404737145914778, 6.920643188701589062786595105132, 7.45086185164202713241881765736, 8.64158583659231655580453627019, 9.1829632514515411632335983345, 10.03778001835528346691703266508, 10.87937352731196619807017014796, 11.99930037847205088028340638995, 13.0985258463904501926606202758, 13.98138328651898738744000678646, 14.0539049772803121997713227373, 15.13686113669684539669276344303, 16.09456912956148670744586460690, 16.859732769401055093895452112617, 17.35405209508137936209628512640, 18.05664110542391384287162015235, 18.78861893944955641022203724562, 19.80411183639351023334105385207, 20.775273132485804959756963354073, 21.54425461151605858614742805390

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