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Properties

Label 1-33e2-1089.137-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.955 + 0.293i$
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  + (0.290 + 0.956i)2-s + (−0.830 + 0.556i)4-s + (−0.988 − 0.151i)5-s + (0.879 − 0.475i)7-s + (−0.774 − 0.633i)8-s + (−0.142 − 0.989i)10-s + (0.999 + 0.0380i)13-s + (0.710 + 0.703i)14-s + (0.380 − 0.924i)16-s + (−0.198 + 0.980i)17-s + (0.993 + 0.113i)19-s + (0.905 − 0.424i)20-s + (−0.723 + 0.690i)23-s + (0.953 + 0.299i)25-s + (0.254 + 0.967i)26-s + ⋯
L(s)  = 1  + (0.290 + 0.956i)2-s + (−0.830 + 0.556i)4-s + (−0.988 − 0.151i)5-s + (0.879 − 0.475i)7-s + (−0.774 − 0.633i)8-s + (−0.142 − 0.989i)10-s + (0.999 + 0.0380i)13-s + (0.710 + 0.703i)14-s + (0.380 − 0.924i)16-s + (−0.198 + 0.980i)17-s + (0.993 + 0.113i)19-s + (0.905 − 0.424i)20-s + (−0.723 + 0.690i)23-s + (0.953 + 0.299i)25-s + (0.254 + 0.967i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.955 + 0.293i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.955 + 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.683954264 + 0.2527862100i\)
\(L(\frac12)\) \(\approx\) \(1.683954264 + 0.2527862100i\)
\(L(1)\) \(\approx\) \(0.9649259973 + 0.4022769084i\)
\(L(1)\) \(\approx\) \(0.9649259973 + 0.4022769084i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.290 + 0.956i)T \)
5 \( 1 + (-0.988 - 0.151i)T \)
7 \( 1 + (0.879 - 0.475i)T \)
13 \( 1 + (0.999 + 0.0380i)T \)
17 \( 1 + (-0.198 + 0.980i)T \)
19 \( 1 + (0.993 + 0.113i)T \)
23 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (0.217 - 0.976i)T \)
31 \( 1 + (-0.969 + 0.244i)T \)
37 \( 1 + (-0.985 - 0.170i)T \)
41 \( 1 + (-0.483 - 0.875i)T \)
43 \( 1 + (-0.888 - 0.458i)T \)
47 \( 1 + (-0.123 - 0.992i)T \)
53 \( 1 + (-0.610 + 0.791i)T \)
59 \( 1 + (0.999 - 0.0190i)T \)
61 \( 1 + (-0.683 - 0.730i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (0.870 - 0.491i)T \)
73 \( 1 + (-0.564 + 0.825i)T \)
79 \( 1 + (0.548 + 0.836i)T \)
83 \( 1 + (0.179 + 0.983i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (0.988 - 0.151i)T \)
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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

−21.02080346824888655044000871484, −20.38971674382616743920248409132, −19.92808536352931659215403012344, −18.80406949917438766756205815055, −18.29937730777802788946747506768, −17.81929790746866763536255743166, −16.32028790487584762104225610366, −15.68842735003241651028480077453, −14.709163020179329969661457373718, −14.178615077349049509930801421079, −13.23262295121836120599836832778, −12.279669221576098620035158799887, −11.58110536418092303649808890518, −11.19231726802192839321187565484, −10.30458883669543225300867398995, −9.15167318318233136153165568132, −8.48483572923123456740439597772, −7.69095073104015594403011129149, −6.47783700190102223023831336684, −5.259283538719941891897443703927, −4.6903814299494321961611224099, −3.647218461050926004148382150706, −2.95015554967227191565103879322, −1.77135728286608034058492032912, −0.791607909071410169086049407759, 0.443930088836849047005589120935, 1.659973263719689614378896203486, 3.62120632462826014498748226335, 3.812830783631045077741208571373, 4.92332521276350086237068906349, 5.68690683251878789299561083581, 6.804991348571833482730350288830, 7.57798472076102238886695571266, 8.24844238218367652274540572012, 8.79506299149684264670293134199, 10.07117964962011627158095062487, 11.13319980555060316614610895048, 11.84083586266518732310297112615, 12.702372666037295560901225676489, 13.71701239279780858734399099411, 14.18001048941019425659337849864, 15.33359330814200798299849717790, 15.57514171191609791368486138031, 16.550048875241823727880983588314, 17.229394692154665703953188432365, 18.07581714949465913627181371245, 18.724911417144427550786921253588, 19.791743432595441121967469790034, 20.5539464140879329091926740065, 21.36374993958920186573037243211

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