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Properties

Label 1-33e2-1089.1004-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.819 - 0.573i$
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  + (−0.953 + 0.299i)2-s + (0.820 − 0.572i)4-s + (0.851 − 0.524i)5-s + (−0.532 − 0.846i)7-s + (−0.610 + 0.791i)8-s + (−0.654 + 0.755i)10-s + (0.797 + 0.603i)13-s + (0.761 + 0.647i)14-s + (0.345 − 0.938i)16-s + (0.254 − 0.967i)17-s + (−0.362 + 0.931i)19-s + (0.398 − 0.917i)20-s + (−0.928 + 0.371i)23-s + (0.449 − 0.893i)25-s + (−0.941 − 0.336i)26-s + ⋯
L(s)  = 1  + (−0.953 + 0.299i)2-s + (0.820 − 0.572i)4-s + (0.851 − 0.524i)5-s + (−0.532 − 0.846i)7-s + (−0.610 + 0.791i)8-s + (−0.654 + 0.755i)10-s + (0.797 + 0.603i)13-s + (0.761 + 0.647i)14-s + (0.345 − 0.938i)16-s + (0.254 − 0.967i)17-s + (−0.362 + 0.931i)19-s + (0.398 − 0.917i)20-s + (−0.928 + 0.371i)23-s + (0.449 − 0.893i)25-s + (−0.941 − 0.336i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.819 - 0.573i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.819 - 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2278124418 - 0.7227104984i\)
\(L(\frac12)\) \(\approx\) \(0.2278124418 - 0.7227104984i\)
\(L(1)\) \(\approx\) \(0.7208020957 - 0.1196831414i\)
\(L(1)\) \(\approx\) \(0.7208020957 - 0.1196831414i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.953 + 0.299i)T \)
5 \( 1 + (0.851 - 0.524i)T \)
7 \( 1 + (-0.532 - 0.846i)T \)
13 \( 1 + (0.797 + 0.603i)T \)
17 \( 1 + (0.254 - 0.967i)T \)
19 \( 1 + (-0.362 + 0.931i)T \)
23 \( 1 + (-0.928 + 0.371i)T \)
29 \( 1 + (-0.548 + 0.836i)T \)
31 \( 1 + (0.483 - 0.875i)T \)
37 \( 1 + (0.974 - 0.226i)T \)
41 \( 1 + (-0.749 + 0.662i)T \)
43 \( 1 + (0.235 - 0.971i)T \)
47 \( 1 + (-0.861 + 0.508i)T \)
53 \( 1 + (0.985 + 0.170i)T \)
59 \( 1 + (0.948 - 0.318i)T \)
61 \( 1 + (-0.217 - 0.976i)T \)
67 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (-0.774 - 0.633i)T \)
73 \( 1 + (0.696 - 0.717i)T \)
79 \( 1 + (-0.432 - 0.901i)T \)
83 \( 1 + (0.0665 - 0.997i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (-0.851 - 0.524i)T \)
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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

−21.447279430579880102367982330513, −20.85400164041209303176062966981, −19.75480089552491772976557634707, −19.19799992273356695819547278170, −18.25744819772679001305175451932, −17.96587010528633158541314862775, −17.07223402844158735052668145980, −16.23669907258051161179715220268, −15.37110165478243502616501548501, −14.80115296113145536576263568979, −13.4234445812516349103614558956, −12.88193757504252983594570232872, −11.91985754456443786075391264537, −11.040878767860127716087112113428, −10.24760093931734252012657874565, −9.70278746149736446078374920050, −8.73299526561554311941018706197, −8.1822239821918796301911157295, −6.90145972709579392591764824284, −6.21392621668778429440367219280, −5.59431910992424544997550342401, −3.88416886477004983633922804042, −2.869501777461802171008111649797, −2.24835874381240343150304358717, −1.17913440057242149529290062545, 0.22019950003866454102124542538, 1.24553334220227225812531268627, 2.03180909343256779433230048508, 3.325364748490832852583775843265, 4.51388021096883850652630953460, 5.74555342256185789820849412282, 6.289787805121686990819101947, 7.209021655990051247855807606, 8.0886976722464151243253011452, 9.0047774956544452548115414886, 9.72922771098577303521022573544, 10.23378172743636592160064872670, 11.20212864753275748645311400277, 12.101685305911750104813030795687, 13.21523932158974648399721643525, 13.88965154048029074751905207736, 14.64073143211589587202842003464, 15.89140713467084890690628915350, 16.51571814753763166868359277760, 16.838524601221976663475311792710, 17.877568663846384857556551399671, 18.4466624992078731146504804999, 19.270685508552942065296768177702, 20.29569750227698243117583769374, 20.587149366159306635596952651505

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