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Properties

Label 1-33e2-1089.590-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.999 + 0.00576i$
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  + (0.345 − 0.938i)2-s + (−0.761 − 0.647i)4-s + (0.595 + 0.803i)5-s + (0.625 + 0.780i)7-s + (−0.870 + 0.491i)8-s + (0.959 − 0.281i)10-s + (0.851 − 0.524i)13-s + (0.948 − 0.318i)14-s + (0.161 + 0.986i)16-s + (0.516 + 0.856i)17-s + (−0.0855 − 0.996i)19-s + (0.0665 − 0.997i)20-s + (−0.0475 + 0.998i)23-s + (−0.290 + 0.956i)25-s + (−0.198 − 0.980i)26-s + ⋯
L(s)  = 1  + (0.345 − 0.938i)2-s + (−0.761 − 0.647i)4-s + (0.595 + 0.803i)5-s + (0.625 + 0.780i)7-s + (−0.870 + 0.491i)8-s + (0.959 − 0.281i)10-s + (0.851 − 0.524i)13-s + (0.948 − 0.318i)14-s + (0.161 + 0.986i)16-s + (0.516 + 0.856i)17-s + (−0.0855 − 0.996i)19-s + (0.0665 − 0.997i)20-s + (−0.0475 + 0.998i)23-s + (−0.290 + 0.956i)25-s + (−0.198 − 0.980i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.00576i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (590, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.999 + 0.00576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.940314145 + 0.005597514761i\)
\(L(\frac12)\) \(\approx\) \(1.940314145 + 0.005597514761i\)
\(L(1)\) \(\approx\) \(1.349360021 - 0.2745148085i\)
\(L(1)\) \(\approx\) \(1.349360021 - 0.2745148085i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.345 - 0.938i)T \)
5 \( 1 + (0.595 + 0.803i)T \)
7 \( 1 + (0.625 + 0.780i)T \)
13 \( 1 + (0.851 - 0.524i)T \)
17 \( 1 + (0.516 + 0.856i)T \)
19 \( 1 + (-0.0855 - 0.996i)T \)
23 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (-0.683 + 0.730i)T \)
31 \( 1 + (-0.432 + 0.901i)T \)
37 \( 1 + (0.610 - 0.791i)T \)
41 \( 1 + (-0.969 - 0.244i)T \)
43 \( 1 + (-0.580 - 0.814i)T \)
47 \( 1 + (0.532 + 0.846i)T \)
53 \( 1 + (-0.774 - 0.633i)T \)
59 \( 1 + (-0.272 + 0.962i)T \)
61 \( 1 + (-0.640 + 0.768i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (0.921 - 0.389i)T \)
73 \( 1 + (0.998 - 0.0570i)T \)
79 \( 1 + (0.217 + 0.976i)T \)
83 \( 1 + (0.964 + 0.263i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (-0.595 + 0.803i)T \)
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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

−21.42707668406622638696294119531, −20.613940015628078657461198871594, −20.40525740931115901843386370281, −18.62607691451170730442312524475, −18.34340404283531143116210751339, −17.16056684007243059328130926642, −16.74508255717398599693330457934, −16.23576557711683105208575345072, −15.1318291696174429201533072276, −14.24008665703532847214912033579, −13.738322113324165348901748772896, −13.06753931303006966711716115073, −12.15486472268522101842668179081, −11.25561188363543232627288412377, −9.99243687851054673027432440473, −9.33935496059525011333050660332, −8.27479147368921227430841474885, −7.87079201582525600126521738738, −6.68737267724609371535664800951, −5.96599990434432502010081856579, −5.03508203484495952756576469760, −4.36239316134823554300831763193, −3.504473130102565058692461396586, −1.936537674044031883950972693954, −0.77351448585515929750308963574, 1.358238896755116619042158648718, 2.08426050813140603479306723859, 3.084375255146958909001686528046, 3.772803043345156305544915696, 5.19308926002546328558423040475, 5.6276622169377258056509311419, 6.5842809948959055310542521224, 7.882939290339487353914586276753, 8.88141372341375948850734174787, 9.51204425082603187841167060752, 10.71768242317614267306439573076, 10.89763539398703996228877777269, 11.88268128031804787133791404779, 12.76979155244939765903587579518, 13.513945988083874111941006704445, 14.28402636710743270140017358887, 15.022024598844825041383818411915, 15.57986449492297881817380440110, 17.128219353146936655525012648349, 17.931626334745940616927399350338, 18.31651612475376074159384769451, 19.11268501925151869152271628360, 19.92644199588692751260005102474, 20.851185730927629658866620299507, 21.609497621742419713317011991139

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