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Properties

Label 1-33e2-1089.592-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.358 - 0.933i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (0.861 + 0.508i)2-s + (0.483 + 0.875i)4-s + (−0.179 + 0.983i)5-s + (−0.830 − 0.556i)7-s + (−0.0285 + 0.999i)8-s + (−0.654 + 0.755i)10-s + (−0.905 − 0.424i)13-s + (−0.432 − 0.901i)14-s + (−0.532 + 0.846i)16-s + (−0.998 + 0.0570i)17-s + (−0.254 − 0.967i)19-s + (−0.948 + 0.318i)20-s + (−0.786 − 0.618i)23-s + (−0.935 − 0.353i)25-s + (−0.564 − 0.825i)26-s + ⋯
L(s)  = 1  + (0.861 + 0.508i)2-s + (0.483 + 0.875i)4-s + (−0.179 + 0.983i)5-s + (−0.830 − 0.556i)7-s + (−0.0285 + 0.999i)8-s + (−0.654 + 0.755i)10-s + (−0.905 − 0.424i)13-s + (−0.432 − 0.901i)14-s + (−0.532 + 0.846i)16-s + (−0.998 + 0.0570i)17-s + (−0.254 − 0.967i)19-s + (−0.948 + 0.318i)20-s + (−0.786 − 0.618i)23-s + (−0.935 − 0.353i)25-s + (−0.564 − 0.825i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.358 - 0.933i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (592, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.358 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02863662735 + 0.04167298968i\)
\(L(\frac12)\) \(\approx\) \(0.02863662735 + 0.04167298968i\)
\(L(1)\) \(\approx\) \(0.9620358490 + 0.4591706915i\)
\(L(1)\) \(\approx\) \(0.9620358490 + 0.4591706915i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.861 + 0.508i)T \)
5 \( 1 + (-0.179 + 0.983i)T \)
7 \( 1 + (-0.830 - 0.556i)T \)
13 \( 1 + (-0.905 - 0.424i)T \)
17 \( 1 + (-0.998 + 0.0570i)T \)
19 \( 1 + (-0.254 - 0.967i)T \)
23 \( 1 + (-0.786 - 0.618i)T \)
29 \( 1 + (0.161 - 0.986i)T \)
31 \( 1 + (-0.290 + 0.956i)T \)
37 \( 1 + (-0.921 - 0.389i)T \)
41 \( 1 + (0.953 - 0.299i)T \)
43 \( 1 + (0.723 + 0.690i)T \)
47 \( 1 + (-0.595 - 0.803i)T \)
53 \( 1 + (-0.466 + 0.884i)T \)
59 \( 1 + (-0.217 - 0.976i)T \)
61 \( 1 + (0.00951 + 0.999i)T \)
67 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (-0.362 + 0.931i)T \)
73 \( 1 + (-0.985 + 0.170i)T \)
79 \( 1 + (0.380 - 0.924i)T \)
83 \( 1 + (0.272 - 0.962i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.179 - 0.983i)T \)
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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

−20.92820128399132193098872864068, −20.16784677173667861111264571535, −19.50721614239156283809418670453, −18.98675669982274927651244014923, −17.86452787170826892862774851996, −16.741907546671627524419435358955, −16.061169380815179557592748086093, −15.451321925245867552631929736998, −14.53833212647040419939995769153, −13.646778179688276694524691103679, −12.84969512649991825858252766897, −12.30818392988217715640987335104, −11.73438460550085074535311066654, −10.6688220196564406983938305044, −9.59233743681722660794262754420, −9.22475777453869438361933257538, −7.98568964102768519047021149990, −6.84824393154118692906184960883, −5.97693110068845401118923285312, −5.211461549597140913033336450957, −4.336046719091072821532576037397, −3.563075742853151919468349322723, −2.40020208361775388355868093311, −1.606759817000227157686349724838, −0.0136985127203343736463224804, 2.31775039427031446435089988734, 2.88661833125116218424214675364, 3.901473498702213756856060998919, 4.60318385399831036558280860327, 5.830320374623317182445424794566, 6.652513161095263257081387180333, 7.12300685716500175976771313570, 7.955195813151221652649468988646, 9.12125028324671520440369305976, 10.27071820701474545319632819426, 10.891774631539250667974100902117, 11.84928345706620690672222958584, 12.69703127898285645581763784459, 13.42486587228875675125830833070, 14.19263225185716284287126555515, 14.87493636648090379975630555828, 15.72352482182826445100582450869, 16.166562357816709016102264535, 17.51781342536473447728262339527, 17.612098843922807491068186803553, 19.07374091001122658164611726534, 19.73569082213132657662419890242, 20.37327083404243167784250145473, 21.641545751037625011342793447964, 22.06678845008791721459874268253

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