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Properties

Label 1-33e2-1089.214-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.0175 - 0.999i$
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  + (0.00951 + 0.999i)2-s + (−0.999 + 0.0190i)4-s + (−0.761 + 0.647i)5-s + (−0.0665 − 0.997i)7-s + (−0.0285 − 0.999i)8-s + (−0.654 − 0.755i)10-s + (0.820 + 0.572i)13-s + (0.997 − 0.0760i)14-s + (0.999 − 0.0380i)16-s + (−0.998 − 0.0570i)17-s + (−0.254 + 0.967i)19-s + (0.749 − 0.662i)20-s + (0.928 + 0.371i)23-s + (0.161 − 0.986i)25-s + (−0.564 + 0.825i)26-s + ⋯
L(s)  = 1  + (0.00951 + 0.999i)2-s + (−0.999 + 0.0190i)4-s + (−0.761 + 0.647i)5-s + (−0.0665 − 0.997i)7-s + (−0.0285 − 0.999i)8-s + (−0.654 − 0.755i)10-s + (0.820 + 0.572i)13-s + (0.997 − 0.0760i)14-s + (0.999 − 0.0380i)16-s + (−0.998 − 0.0570i)17-s + (−0.254 + 0.967i)19-s + (0.749 − 0.662i)20-s + (0.928 + 0.371i)23-s + (0.161 − 0.986i)25-s + (−0.564 + 0.825i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0175 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0175 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.0175 - 0.999i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.0175 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06241905322 - 0.06133018437i\)
\(L(\frac12)\) \(\approx\) \(0.06241905322 - 0.06133018437i\)
\(L(1)\) \(\approx\) \(0.5901350849 + 0.3175974353i\)
\(L(1)\) \(\approx\) \(0.5901350849 + 0.3175974353i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.00951 + 0.999i)T \)
5 \( 1 + (-0.761 + 0.647i)T \)
7 \( 1 + (-0.0665 - 0.997i)T \)
13 \( 1 + (0.820 + 0.572i)T \)
17 \( 1 + (-0.998 - 0.0570i)T \)
19 \( 1 + (-0.254 + 0.967i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (-0.935 - 0.353i)T \)
31 \( 1 + (-0.683 + 0.730i)T \)
37 \( 1 + (-0.921 + 0.389i)T \)
41 \( 1 + (-0.217 - 0.976i)T \)
43 \( 1 + (0.235 + 0.971i)T \)
47 \( 1 + (-0.398 - 0.917i)T \)
53 \( 1 + (-0.466 - 0.884i)T \)
59 \( 1 + (0.953 - 0.299i)T \)
61 \( 1 + (0.861 + 0.508i)T \)
67 \( 1 + (-0.995 - 0.0950i)T \)
71 \( 1 + (-0.362 - 0.931i)T \)
73 \( 1 + (-0.985 - 0.170i)T \)
79 \( 1 + (-0.991 - 0.132i)T \)
83 \( 1 + (-0.969 - 0.244i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.761 - 0.647i)T \)
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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

−21.58720362389483212532750864745, −20.6025598547048048784785924525, −20.27407429870646674053038248377, −19.26896401733442292732198553841, −18.80298042419066149786152365927, −17.91958331453039780970391630478, −17.1854734208908591518409915116, −16.069975782082083511096171393965, −15.34053680688941619344322678188, −14.66432852417678424377486502591, −13.13984020974060511395213202563, −13.069288751818711661973641417165, −12.12213707941294901768876036527, −11.22602166362747463285637787817, −10.90926429482621070177490937124, −9.504782125285060434328854934, −8.78502759272290058146324455982, −8.45348703752765039626112433670, −7.21091209133628422442953087178, −5.84059677745819349086310463955, −5.04707668653653947241674098383, −4.21535190852462686279845711966, −3.28099542823311055837451056438, −2.3820950254217360745811590733, −1.259011320592431891885692070849, 0.03978649787396066321797722216, 1.5630360744081701421418949480, 3.38890231738293582579612393301, 3.89713109190837158613655184257, 4.744624039119012192514825884564, 5.957221774802336349764517152471, 6.89870206340087506179306394796, 7.21997142052043601715670646343, 8.24521002808459424472413898961, 8.94255013023913111357195504542, 10.08212750713945059506053084758, 10.84089827809511423203438050143, 11.64775580039191359506962812210, 12.90131073412632967339922527552, 13.53139256819067337358561859427, 14.377246071024754863447174292426, 14.962383984841733057163328433692, 15.91083287741472087991489979454, 16.38030983410025491460284261257, 17.2788902035622805092067629912, 18.02917755172383399864147883040, 18.90420887079001397039606678846, 19.39360513266377143837231436375, 20.46428048418879472965924726387, 21.3268137632252014292433422231

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