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Properties

Label 1-33e2-1089.1075-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  + (0.830 − 0.556i)2-s + (0.380 − 0.924i)4-s + (0.953 + 0.299i)5-s + (−0.548 + 0.836i)7-s + (−0.198 − 0.980i)8-s + (0.959 − 0.281i)10-s + (−0.997 − 0.0760i)13-s + (0.00951 + 0.999i)14-s + (−0.710 − 0.703i)16-s + (0.921 + 0.389i)17-s + (−0.974 − 0.226i)19-s + (0.640 − 0.768i)20-s + (0.0475 − 0.998i)23-s + (0.820 + 0.572i)25-s + (−0.870 + 0.491i)26-s + ⋯
L(s)  = 1  + (0.830 − 0.556i)2-s + (0.380 − 0.924i)4-s + (0.953 + 0.299i)5-s + (−0.548 + 0.836i)7-s + (−0.198 − 0.980i)8-s + (0.959 − 0.281i)10-s + (−0.997 − 0.0760i)13-s + (0.00951 + 0.999i)14-s + (−0.710 − 0.703i)16-s + (0.921 + 0.389i)17-s + (−0.974 − 0.226i)19-s + (0.640 − 0.768i)20-s + (0.0475 − 0.998i)23-s + (0.820 + 0.572i)25-s + (−0.870 + 0.491i)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: $\chi_{1089} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.00576 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.664479631 - 2.649150606i\)
\(L(\frac12)\) \(\approx\) \(2.664479631 - 2.649150606i\)
\(L(1)\) \(\approx\) \(1.718571332 - 0.6702122925i\)
\(L(1)\) \(\approx\) \(1.718571332 - 0.6702122925i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.830 - 0.556i)T \)
5 \( 1 + (0.953 + 0.299i)T \)
7 \( 1 + (-0.548 + 0.836i)T \)
13 \( 1 + (-0.997 - 0.0760i)T \)
17 \( 1 + (0.921 + 0.389i)T \)
19 \( 1 + (-0.974 - 0.226i)T \)
23 \( 1 + (0.0475 - 0.998i)T \)
29 \( 1 + (0.905 + 0.424i)T \)
31 \( 1 + (0.879 - 0.475i)T \)
37 \( 1 + (0.941 + 0.336i)T \)
41 \( 1 + (0.532 - 0.846i)T \)
43 \( 1 + (-0.580 - 0.814i)T \)
47 \( 1 + (-0.969 + 0.244i)T \)
53 \( 1 + (-0.254 - 0.967i)T \)
59 \( 1 + (0.999 - 0.0380i)T \)
61 \( 1 + (0.0665 - 0.997i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (0.362 + 0.931i)T \)
79 \( 1 + (0.398 - 0.917i)T \)
83 \( 1 + (0.935 - 0.353i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.953 - 0.299i)T \)
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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

−21.37696463526451232396785349961, −21.06617030349179951014616243312, −19.97258742254723007678793659278, −19.35874529078437139636788240439, −17.976449169968348673659857289400, −17.29246044423841499773938048990, −16.69892626814822172494081494108, −16.13380927931963724491304842805, −14.992240647275883787097081711408, −14.269691523025193067445549437201, −13.65393217369978501305534213804, −12.91720649164093565166030769796, −12.30472336400220433704290570919, −11.279612081419020287685885944184, −10.08438653507712517984694156316, −9.62513151338981731552890844987, −8.36287750328944071476704021491, −7.513850519183911689071285249618, −6.643309022099055764445076080246, −5.99010497561778397788898347693, −5.01065318504872225758379696230, −4.325211097105717845721465029238, −3.17992153225864025347384030879, −2.36323043223496801454277434109, −1.031183665651091787476955582101, 0.588834549151473226601866989358, 2.02439945038704442131956063278, 2.53799149456804370459498391114, 3.3987004844044875242748508561, 4.69867526741417652590976424302, 5.39649671665090190528290345314, 6.316806708318375700424468763342, 6.74299261006277054874489350504, 8.26579223587024530638373300819, 9.39365858986839126046451753991, 10.00238047766829189244341293393, 10.626033631894794036608530648939, 11.74098765306023565155247954363, 12.56902053180025423512369112871, 12.94036796085641247897868365396, 14.01906945053673964713706887359, 14.67562682059993742119236710190, 15.22217742203496955667484068920, 16.30329586688558442450180437217, 17.14948758695501677522919599387, 18.14499567277908485169477204351, 18.9887628445316296643427557719, 19.392699521952652960477363715560, 20.498240669497800876589487949455, 21.29130533558585149495540942938

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