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Properties

Label 1-33e2-1089.5-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.00576 - 0.999i$
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  + (−0.999 − 0.0380i)2-s + (0.997 + 0.0760i)4-s + (0.948 − 0.318i)5-s + (0.964 + 0.263i)7-s + (−0.993 − 0.113i)8-s + (−0.959 + 0.281i)10-s + (−0.761 − 0.647i)13-s + (−0.953 − 0.299i)14-s + (0.988 + 0.151i)16-s + (−0.974 + 0.226i)17-s + (0.516 − 0.856i)19-s + (0.969 − 0.244i)20-s + (−0.0475 + 0.998i)23-s + (0.797 − 0.603i)25-s + (0.736 + 0.676i)26-s + ⋯
L(s)  = 1  + (−0.999 − 0.0380i)2-s + (0.997 + 0.0760i)4-s + (0.948 − 0.318i)5-s + (0.964 + 0.263i)7-s + (−0.993 − 0.113i)8-s + (−0.959 + 0.281i)10-s + (−0.761 − 0.647i)13-s + (−0.953 − 0.299i)14-s + (0.988 + 0.151i)16-s + (−0.974 + 0.226i)17-s + (0.516 − 0.856i)19-s + (0.969 − 0.244i)20-s + (−0.0475 + 0.998i)23-s + (0.797 − 0.603i)25-s + (0.736 + 0.676i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00576 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00576 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.00576 - 0.999i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.00576 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9847779034 - 0.9791123751i\)
\(L(\frac12)\) \(\approx\) \(0.9847779034 - 0.9791123751i\)
\(L(1)\) \(\approx\) \(0.8498087395 - 0.1411426800i\)
\(L(1)\) \(\approx\) \(0.8498087395 - 0.1411426800i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.999 - 0.0380i)T \)
5 \( 1 + (0.948 - 0.318i)T \)
7 \( 1 + (0.964 + 0.263i)T \)
13 \( 1 + (-0.761 - 0.647i)T \)
17 \( 1 + (-0.974 + 0.226i)T \)
19 \( 1 + (0.516 - 0.856i)T \)
23 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (-0.123 + 0.992i)T \)
31 \( 1 + (-0.991 - 0.132i)T \)
37 \( 1 + (-0.0285 + 0.999i)T \)
41 \( 1 + (-0.640 - 0.768i)T \)
43 \( 1 + (0.580 + 0.814i)T \)
47 \( 1 + (0.0665 - 0.997i)T \)
53 \( 1 + (0.362 - 0.931i)T \)
59 \( 1 + (-0.345 - 0.938i)T \)
61 \( 1 + (-0.532 - 0.846i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (-0.0855 - 0.996i)T \)
73 \( 1 + (0.774 - 0.633i)T \)
79 \( 1 + (0.861 - 0.508i)T \)
83 \( 1 + (-0.548 + 0.836i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (-0.948 - 0.318i)T \)
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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

−21.28084222181702142046861487720, −20.56131758890841242239959335621, −19.94685137910396496620909842654, −18.82273526617899299867443403189, −18.28981704711500728859247530609, −17.57874782551355223099591392317, −16.988153991964735488377549260304, −16.28992012293552322580081546563, −15.14687819947521574667394512744, −14.44836405895370442054275236998, −13.84197406881605411091167708445, −12.59490986762418684623291625007, −11.68404285127439023204310344833, −10.9049006988253219426644234829, −10.25755440463136443979228538846, −9.400065156936192813094186052581, −8.75096691730761891219721283979, −7.68755404409581059757381021358, −7.03779850889561177969303484976, −6.12899658790034292418781711575, −5.24175162089652109492485702106, −4.0907809269046374888635948962, −2.538109450393043003041889440681, −2.032497025103890335536177221062, −1.02702826836311965877579887037, 0.40212446061323633928894359656, 1.60701969459647768056024303925, 2.17035498900878868518328779935, 3.254287586593734669425794315910, 4.92741911196742916068373159452, 5.45550710155294495373291344492, 6.58301565263763640523112026820, 7.42155048240003743592529795582, 8.314776645125467321623394650990, 9.087664977254270782693164614810, 9.68802667825519753911119397954, 10.66934132535749635373356431733, 11.27860654312799218947327714525, 12.227302804767443578278726286950, 13.08064455233734201102649407430, 14.05641407268902018430915551919, 15.01930026067774750339360565076, 15.57922531968039745584698613655, 16.718509908461325898489716831997, 17.348657426800625076060985297763, 17.91781442910495836679726954979, 18.37151092048985104671208258421, 19.646115946681365966990950136394, 20.16807621813912273156533625893, 20.8740751258715778331887591072

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