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Properties

Label 1-33e2-1089.382-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.514 + 0.857i$
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  + (0.851 + 0.524i)2-s + (0.449 + 0.893i)4-s + (−0.999 − 0.0190i)5-s + (−0.749 − 0.662i)7-s + (−0.0855 + 0.996i)8-s + (−0.841 − 0.540i)10-s + (0.710 − 0.703i)13-s + (−0.290 − 0.956i)14-s + (−0.595 + 0.803i)16-s + (0.985 − 0.170i)17-s + (−0.696 − 0.717i)19-s + (−0.432 − 0.901i)20-s + (−0.995 + 0.0950i)23-s + (0.999 + 0.0380i)25-s + (0.974 − 0.226i)26-s + ⋯
L(s)  = 1  + (0.851 + 0.524i)2-s + (0.449 + 0.893i)4-s + (−0.999 − 0.0190i)5-s + (−0.749 − 0.662i)7-s + (−0.0855 + 0.996i)8-s + (−0.841 − 0.540i)10-s + (0.710 − 0.703i)13-s + (−0.290 − 0.956i)14-s + (−0.595 + 0.803i)16-s + (0.985 − 0.170i)17-s + (−0.696 − 0.717i)19-s + (−0.432 − 0.901i)20-s + (−0.995 + 0.0950i)23-s + (0.999 + 0.0380i)25-s + (0.974 − 0.226i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.514 + 0.857i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (382, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8942965432 + 1.579866122i\)
\(L(\frac12)\) \(\approx\) \(0.8942965432 + 1.579866122i\)
\(L(1)\) \(\approx\) \(1.196920971 + 0.4340418436i\)
\(L(1)\) \(\approx\) \(1.196920971 + 0.4340418436i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.851 + 0.524i)T \)
5 \( 1 + (-0.999 - 0.0190i)T \)
7 \( 1 + (-0.749 - 0.662i)T \)
13 \( 1 + (0.710 - 0.703i)T \)
17 \( 1 + (0.985 - 0.170i)T \)
19 \( 1 + (-0.696 - 0.717i)T \)
23 \( 1 + (-0.995 + 0.0950i)T \)
29 \( 1 + (0.532 + 0.846i)T \)
31 \( 1 + (-0.935 - 0.353i)T \)
37 \( 1 + (-0.362 - 0.931i)T \)
41 \( 1 + (0.991 + 0.132i)T \)
43 \( 1 + (0.327 + 0.945i)T \)
47 \( 1 + (-0.179 + 0.983i)T \)
53 \( 1 + (0.993 - 0.113i)T \)
59 \( 1 + (0.380 - 0.924i)T \)
61 \( 1 + (-0.879 - 0.475i)T \)
67 \( 1 + (0.723 + 0.690i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (0.870 - 0.491i)T \)
79 \( 1 + (-0.123 + 0.992i)T \)
83 \( 1 + (0.217 + 0.976i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (-0.999 + 0.0190i)T \)
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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

−21.152548989643753862638093610644, −20.19110182726646512481948077345, −19.49814057880692364670308402405, −18.809113908090577318437543142, −18.42346490005507954900352323567, −16.70536886475254897376393653283, −16.106528035056681231220989294944, −15.43845428684467951187612054443, −14.695030413004081159354713930724, −13.88767843349526436375881566685, −12.94684632299850096807106583312, −12.0475838690435216681591023968, −11.92552164904425414721898497067, −10.75003699617794255725656591568, −10.04930665476175612910035014972, −9.00551905591987318436197599183, −8.10041450229182680253265980916, −6.95091876794385719866053479189, −6.15570909281098428516571175001, −5.403231330270832290776527812267, −4.09540167884182383201228790310, −3.729432950455331388331083911099, −2.71594106521561334740536926085, −1.64672671277672417451195460553, −0.33473552309304175107631212862, 0.87636965026338645349989242142, 2.62227797810628353924023527712, 3.57641503354469423301181921439, 3.98574026533847010109696280859, 5.077342327094807875920938850737, 6.05811313560444312395750283933, 6.87918931142257428887743896182, 7.67708653340786503571972502691, 8.271066939102532801617645169077, 9.40799256846510195909481742319, 10.72723537322803681240380761735, 11.185992873573353759955862427099, 12.52953304439840724788175316749, 12.61093440661572741195043154037, 13.71755472913214885765940365419, 14.46410607128235417937782125792, 15.30037068928640878434988550249, 16.15966709963252961880726988216, 16.32559885662770372142054181858, 17.43848382155344682420936023179, 18.324911197415562287154865325476, 19.50084718116929243588648084187, 19.96506780941276388238721376884, 20.75978922340579367694057553818, 21.62808313461873379411931855249

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