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Properties

Label 1-33e2-1089.1010-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.336 - 0.941i$
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  + (0.969 + 0.244i)2-s + (0.879 + 0.475i)4-s + (−0.861 + 0.508i)5-s + (0.161 − 0.986i)7-s + (0.736 + 0.676i)8-s + (−0.959 + 0.281i)10-s + (−0.991 + 0.132i)13-s + (0.398 − 0.917i)14-s + (0.548 + 0.836i)16-s + (−0.0855 + 0.996i)17-s + (−0.921 + 0.389i)19-s + (−0.999 + 0.0380i)20-s + (0.888 − 0.458i)23-s + (0.483 − 0.875i)25-s + (−0.993 − 0.113i)26-s + ⋯
L(s)  = 1  + (0.969 + 0.244i)2-s + (0.879 + 0.475i)4-s + (−0.861 + 0.508i)5-s + (0.161 − 0.986i)7-s + (0.736 + 0.676i)8-s + (−0.959 + 0.281i)10-s + (−0.991 + 0.132i)13-s + (0.398 − 0.917i)14-s + (0.548 + 0.836i)16-s + (−0.0855 + 0.996i)17-s + (−0.921 + 0.389i)19-s + (−0.999 + 0.0380i)20-s + (0.888 − 0.458i)23-s + (0.483 − 0.875i)25-s + (−0.993 − 0.113i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.336 - 0.941i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1010, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.336 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5527189328 - 0.7845376420i\)
\(L(\frac12)\) \(\approx\) \(0.5527189328 - 0.7845376420i\)
\(L(1)\) \(\approx\) \(1.370182908 + 0.2004920231i\)
\(L(1)\) \(\approx\) \(1.370182908 + 0.2004920231i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.969 + 0.244i)T \)
5 \( 1 + (-0.861 + 0.508i)T \)
7 \( 1 + (0.161 - 0.986i)T \)
13 \( 1 + (-0.991 + 0.132i)T \)
17 \( 1 + (-0.0855 + 0.996i)T \)
19 \( 1 + (-0.921 + 0.389i)T \)
23 \( 1 + (0.888 - 0.458i)T \)
29 \( 1 + (0.999 + 0.0190i)T \)
31 \( 1 + (-0.761 + 0.647i)T \)
37 \( 1 + (-0.564 - 0.825i)T \)
41 \( 1 + (0.830 - 0.556i)T \)
43 \( 1 + (-0.995 + 0.0950i)T \)
47 \( 1 + (-0.345 - 0.938i)T \)
53 \( 1 + (0.998 + 0.0570i)T \)
59 \( 1 + (0.0665 - 0.997i)T \)
61 \( 1 + (0.272 - 0.962i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (-0.974 - 0.226i)T \)
73 \( 1 + (-0.254 + 0.967i)T \)
79 \( 1 + (-0.948 + 0.318i)T \)
83 \( 1 + (-0.988 + 0.151i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (0.861 + 0.508i)T \)
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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

−21.45296044851771134892930098213, −20.87317543456195698032124859332, −19.944723293765125034518747933855, −19.37860179016648759882066180552, −18.70527484757047868649284916478, −17.52258689727915973160677929209, −16.50142301312401754176281522964, −15.84750901579448494675170269802, −14.980978687523111763990175043380, −14.74743192367937826153098859917, −13.38947594367315027198853234019, −12.79941903793293435886740690592, −11.84874101000690094184905665432, −11.679395088988615440959678870241, −10.597165084745074758139275333769, −9.48305720176269080698650367028, −8.644460468236681682263050499713, −7.57801477247698214714580646147, −6.85352857659372670970964473566, −5.6926808637421898772190588582, −4.89530904095226005015337131029, −4.38749681322009940841625455538, −3.08454305868715726286363492202, −2.47771171183131581748856927591, −1.19774319046210817900932873334, 0.13137736393196110097164286460, 1.72121060707868513774734258697, 2.84597937815106186802232381285, 3.79805897333220383347470411356, 4.34780774017051525099736719129, 5.25236176589537261473422757444, 6.58531969279217108715411186885, 7.00689750478797943283386021306, 7.84161001785212773557105308463, 8.620167799869724571023776857739, 10.35324468448522287919211323819, 10.67542176039035972953512071376, 11.61062493136422805364424696582, 12.46757047919987120372655195968, 13.034428863756631138294159838389, 14.25772414434461312780397962827, 14.58384686606272773546337506965, 15.32194394470601783631240382926, 16.25891534599471763239381832202, 16.93767715797863764671888939535, 17.621089333557309321554895939579, 18.98831576472135156781482001397, 19.67128009322783823268428720213, 20.147486050144507498212341525778, 21.26690359452891435060885864719

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