# Properties

 Label 1-33e2-1089.7-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 $0$

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## 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: $\chi_{1089} (7, \cdot )$ 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

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