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Properties

Label 1-33e2-1089.1066-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.999 + 0.0187i$
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  + (−0.981 + 0.189i)2-s + (0.928 − 0.371i)4-s + (0.0475 − 0.998i)5-s + (−0.235 + 0.971i)7-s + (−0.841 + 0.540i)8-s + (0.142 + 0.989i)10-s + (−0.928 + 0.371i)13-s + (0.0475 − 0.998i)14-s + (0.723 − 0.690i)16-s + (−0.415 − 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.327 − 0.945i)20-s + (0.235 + 0.971i)23-s + (−0.995 − 0.0950i)25-s + (0.841 − 0.540i)26-s + ⋯
L(s)  = 1  + (−0.981 + 0.189i)2-s + (0.928 − 0.371i)4-s + (0.0475 − 0.998i)5-s + (−0.235 + 0.971i)7-s + (−0.841 + 0.540i)8-s + (0.142 + 0.989i)10-s + (−0.928 + 0.371i)13-s + (0.0475 − 0.998i)14-s + (0.723 − 0.690i)16-s + (−0.415 − 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.327 − 0.945i)20-s + (0.235 + 0.971i)23-s + (−0.995 − 0.0950i)25-s + (0.841 − 0.540i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.0187i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1066, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.999 + 0.0187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7339594470 + 0.006881611396i\)
\(L(\frac12)\) \(\approx\) \(0.7339594470 + 0.006881611396i\)
\(L(1)\) \(\approx\) \(0.5864136146 + 0.005807667497i\)
\(L(1)\) \(\approx\) \(0.5864136146 + 0.005807667497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.981 + 0.189i)T \)
5 \( 1 + (0.0475 - 0.998i)T \)
7 \( 1 + (-0.235 + 0.971i)T \)
13 \( 1 + (-0.928 + 0.371i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (0.235 + 0.971i)T \)
29 \( 1 + (-0.580 - 0.814i)T \)
31 \( 1 + (-0.786 + 0.618i)T \)
37 \( 1 + (-0.142 - 0.989i)T \)
41 \( 1 + (0.327 - 0.945i)T \)
43 \( 1 + (-0.0475 - 0.998i)T \)
47 \( 1 + (-0.327 - 0.945i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (0.981 + 0.189i)T \)
61 \( 1 + (0.327 + 0.945i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (0.888 - 0.458i)T \)
83 \( 1 + (-0.235 + 0.971i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.0475 + 0.998i)T \)
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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

−21.1309436483552111147598226480, −20.11207652733798749641487846378, −19.67798232419277928622587291671, −18.944637233783078118774278098219, −18.13383458362872143586653321631, −17.35613545605161270929322353716, −16.88939636290500605195577338884, −15.91578600588451893430227930003, −14.936588371076150189226486831949, −14.51399948071533740751144926559, −13.1517096354312482485726824707, −12.584261110756100218038814667, −11.15533945308184010394839261547, −10.96915845166684086639997228932, −10.05365713261958695242133759201, −9.47850476414537035342624113788, −8.2729609589017513784368759611, −7.52350084133861709353833026367, −6.773370314499454879648261687361, −6.23246172400323487684083334590, −4.65766879567294840541421573306, −3.50845224049405244167071001622, −2.74039446494301828729298803004, −1.77176793340653210375200325549, −0.43521085252073846709070986017, 0.441142840752585206362267499566, 1.79976606033126954124171367460, 2.38236697650381924689619508018, 3.79075922971667375717921196736, 5.23246961027032487183729597446, 5.61502950237471579306046450201, 6.81121750094946390150636440944, 7.65203571322047100529785472992, 8.54452121602484556867160337643, 9.30461199400435563688580308302, 9.62204072221680622139138568442, 10.81689170463672062889055579759, 11.94294933097888593014228242238, 12.1609331903376216132568586309, 13.26753590347991739096702485978, 14.42033319716946343196371994123, 15.261137098650944350577749846500, 15.99140731681720689514047259374, 16.58114407712340607044408450004, 17.37996498579648350749981496408, 18.05446940355950899805252292892, 19.03543000385095610656645974366, 19.4668055500983474635786086879, 20.385966179829678395184134370142, 21.06652865269110278401793115800

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