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Properties

Label 1-33e2-1089.104-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.874 - 0.484i$
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  + (0.179 − 0.983i)2-s + (−0.935 − 0.353i)4-s + (0.683 + 0.730i)5-s + (0.953 − 0.299i)7-s + (−0.516 + 0.856i)8-s + (0.841 − 0.540i)10-s + (0.548 + 0.836i)13-s + (−0.123 − 0.992i)14-s + (0.749 + 0.662i)16-s + (0.466 − 0.884i)17-s + (−0.985 + 0.170i)19-s + (−0.380 − 0.924i)20-s + (−0.580 + 0.814i)23-s + (−0.0665 + 0.997i)25-s + (0.921 − 0.389i)26-s + ⋯
L(s)  = 1  + (0.179 − 0.983i)2-s + (−0.935 − 0.353i)4-s + (0.683 + 0.730i)5-s + (0.953 − 0.299i)7-s + (−0.516 + 0.856i)8-s + (0.841 − 0.540i)10-s + (0.548 + 0.836i)13-s + (−0.123 − 0.992i)14-s + (0.749 + 0.662i)16-s + (0.466 − 0.884i)17-s + (−0.985 + 0.170i)19-s + (−0.380 − 0.924i)20-s + (−0.580 + 0.814i)23-s + (−0.0665 + 0.997i)25-s + (0.921 − 0.389i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.874 - 0.484i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.874 - 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.797287472 - 0.7227820556i\)
\(L(\frac12)\) \(\approx\) \(2.797287472 - 0.7227820556i\)
\(L(1)\) \(\approx\) \(1.324519538 - 0.4574810508i\)
\(L(1)\) \(\approx\) \(1.324519538 - 0.4574810508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.179 - 0.983i)T \)
5 \( 1 + (0.683 + 0.730i)T \)
7 \( 1 + (0.953 - 0.299i)T \)
13 \( 1 + (0.548 + 0.836i)T \)
17 \( 1 + (0.466 - 0.884i)T \)
19 \( 1 + (-0.985 + 0.170i)T \)
23 \( 1 + (-0.580 + 0.814i)T \)
29 \( 1 + (0.830 - 0.556i)T \)
31 \( 1 + (0.988 - 0.151i)T \)
37 \( 1 + (-0.254 - 0.967i)T \)
41 \( 1 + (0.851 - 0.524i)T \)
43 \( 1 + (0.981 - 0.189i)T \)
47 \( 1 + (-0.997 + 0.0760i)T \)
53 \( 1 + (-0.198 - 0.980i)T \)
59 \( 1 + (-0.879 + 0.475i)T \)
61 \( 1 + (-0.761 + 0.647i)T \)
67 \( 1 + (0.235 + 0.971i)T \)
71 \( 1 + (-0.696 + 0.717i)T \)
73 \( 1 + (0.993 - 0.113i)T \)
79 \( 1 + (0.820 + 0.572i)T \)
83 \( 1 + (-0.00951 + 0.999i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (-0.683 + 0.730i)T \)
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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

−21.32343849155837399297697661710, −20.8705503689762291952546257527, −19.788276908385138833693780230527, −18.65120181404473582488176746611, −17.9059887294217301432841780138, −17.36799295551859957256272709163, −16.72604707793192032509162016551, −15.80079920896144929047811801775, −15.075572534049461486124650665, −14.28839411250688700944919750797, −13.63761172234186065913910116634, −12.64996819654270005702503675626, −12.28152101771762447518425065195, −10.80471407285210913304866173192, −10.03565380222688497039798868860, −8.920932238006314362993206218252, −8.3177662341359492885344323703, −7.8488792492516940993731832827, −6.28112298415445557882037631035, −6.00909772578687410641825498107, −4.845886916964226372230427137296, −4.4633319087970432580556247939, −3.06744102550241831554405808534, −1.70971568005126326423756680680, −0.687930371853452446209107913319, 0.9145058051634615565264261786, 1.886959303450625642086373636845, 2.56313531713595495997873737710, 3.74626472110355037597293614068, 4.508399906396964895120918413587, 5.51940662725985741469573466874, 6.35456639828639144133806216590, 7.51552242252566082083437262906, 8.49570403610663920321565817353, 9.42162631424184050949501262103, 10.15427625390284349465819348214, 10.959738391216860704117545323744, 11.49372680327943227951308150364, 12.3433216649139899572105627140, 13.52447838915191894960891708689, 14.0033480423039307731740976362, 14.48778525917188603495145487728, 15.51074187837567399430804987138, 16.76990968621449471669993005668, 17.75402036474102446903517316917, 17.97748858474801418686160106278, 19.02242372051749778076635318908, 19.48417250047607506165602653075, 20.83996122341604081321673040812, 21.05750152502010728676228714310

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