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Properties

Label 1-33e2-1089.47-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.604 + 0.796i$
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  + (0.935 + 0.353i)2-s + (0.749 + 0.662i)4-s + (0.0665 − 0.997i)5-s + (0.820 − 0.572i)7-s + (0.466 + 0.884i)8-s + (0.415 − 0.909i)10-s + (−0.398 + 0.917i)13-s + (0.969 − 0.244i)14-s + (0.123 + 0.992i)16-s + (0.564 + 0.825i)17-s + (0.941 − 0.336i)19-s + (0.710 − 0.703i)20-s + (0.327 + 0.945i)23-s + (−0.991 − 0.132i)25-s + (−0.696 + 0.717i)26-s + ⋯
L(s)  = 1  + (0.935 + 0.353i)2-s + (0.749 + 0.662i)4-s + (0.0665 − 0.997i)5-s + (0.820 − 0.572i)7-s + (0.466 + 0.884i)8-s + (0.415 − 0.909i)10-s + (−0.398 + 0.917i)13-s + (0.969 − 0.244i)14-s + (0.123 + 0.992i)16-s + (0.564 + 0.825i)17-s + (0.941 − 0.336i)19-s + (0.710 − 0.703i)20-s + (0.327 + 0.945i)23-s + (−0.991 − 0.132i)25-s + (−0.696 + 0.717i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.604 + 0.796i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.604 + 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.414780580 + 2.190405022i\)
\(L(\frac12)\) \(\approx\) \(4.414780580 + 2.190405022i\)
\(L(1)\) \(\approx\) \(2.150126363 + 0.4450027574i\)
\(L(1)\) \(\approx\) \(2.150126363 + 0.4450027574i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.935 + 0.353i)T \)
5 \( 1 + (0.0665 - 0.997i)T \)
7 \( 1 + (0.820 - 0.572i)T \)
13 \( 1 + (-0.398 + 0.917i)T \)
17 \( 1 + (0.564 + 0.825i)T \)
19 \( 1 + (0.941 - 0.336i)T \)
23 \( 1 + (0.327 + 0.945i)T \)
29 \( 1 + (-0.380 + 0.924i)T \)
31 \( 1 + (0.953 - 0.299i)T \)
37 \( 1 + (-0.870 + 0.491i)T \)
41 \( 1 + (-0.449 + 0.893i)T \)
43 \( 1 + (0.928 - 0.371i)T \)
47 \( 1 + (-0.988 + 0.151i)T \)
53 \( 1 + (0.921 - 0.389i)T \)
59 \( 1 + (-0.548 + 0.836i)T \)
61 \( 1 + (0.161 - 0.986i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (0.0285 + 0.999i)T \)
73 \( 1 + (0.974 - 0.226i)T \)
79 \( 1 + (0.345 + 0.938i)T \)
83 \( 1 + (0.999 + 0.0190i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (-0.0665 - 0.997i)T \)
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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

−21.01710185360919600646173874937, −20.74585492228489945680999486080, −19.64123591370137802095317288538, −18.83399467272501806206898627754, −18.221004709315206085693898474460, −17.38857511855214297438458748684, −16.12870487581396146182649635172, −15.35667435389383097489203664653, −14.74959970379430890996015645048, −14.12474752056257582331445224298, −13.43349087260795925826547468404, −12.14722167930405074484170894388, −11.88521796389606324499296711056, −10.8451236909587235583395862339, −10.294879628791717900047623874518, −9.36061727683947008515123403382, −7.920867842875494188436432050907, −7.31424223452408575932870648606, −6.2580396114162640549149267045, −5.45082026007609122790183906783, −4.78216248106066549886758466405, −3.52764742266705745279261115637, −2.781721145549750014198164810217, −2.055503999927967490828624385042, −0.734972329598776430465857767783, 1.17600192452001380665616202245, 1.88904104340329509684309077380, 3.32309104332414091448564509493, 4.21482463146383054375119958422, 4.95320093387708072806373876019, 5.5355326432037483894783355445, 6.7200894031360497623685826332, 7.57439292144873478697776760601, 8.24830534545780009692938150919, 9.21739710383710199783506349827, 10.30890103356803487772159360880, 11.46385768330711695513704319222, 11.87613081339240145895089579682, 12.815224244562988580903953178534, 13.60429563780341068627676722500, 14.1612823031169610538057303950, 15.02075886024835657180810683623, 15.86190765016789775300459187530, 16.75562528680210862709524518468, 17.09786248448181959170620293444, 17.95600637365515520685764861410, 19.36606927498082306717470739954, 20.00754703814736916391399678794, 20.86987499032004637453559586317, 21.264903111338263030670549478750

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