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Properties

Label 1-33e2-1089.97-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  + (0.999 − 0.0380i)2-s + (0.997 − 0.0760i)4-s + (−0.948 − 0.318i)5-s + (0.964 − 0.263i)7-s + (0.993 − 0.113i)8-s + (−0.959 − 0.281i)10-s + (−0.761 + 0.647i)13-s + (0.953 − 0.299i)14-s + (0.988 − 0.151i)16-s + (0.974 + 0.226i)17-s + (0.516 + 0.856i)19-s + (−0.969 − 0.244i)20-s + (0.0475 + 0.998i)23-s + (0.797 + 0.603i)25-s + (−0.736 + 0.676i)26-s + ⋯
L(s)  = 1  + (0.999 − 0.0380i)2-s + (0.997 − 0.0760i)4-s + (−0.948 − 0.318i)5-s + (0.964 − 0.263i)7-s + (0.993 − 0.113i)8-s + (−0.959 − 0.281i)10-s + (−0.761 + 0.647i)13-s + (0.953 − 0.299i)14-s + (0.988 − 0.151i)16-s + (0.974 + 0.226i)17-s + (0.516 + 0.856i)19-s + (−0.969 − 0.244i)20-s + (0.0475 + 0.998i)23-s + (0.797 + 0.603i)25-s + (−0.736 + 0.676i)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: $\chi_{1089} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.999 + 0.00576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.927898627 + 0.008446547599i\)
\(L(\frac12)\) \(\approx\) \(2.927898627 + 0.008446547599i\)
\(L(1)\) \(\approx\) \(1.936416666 - 0.06021066101i\)
\(L(1)\) \(\approx\) \(1.936416666 - 0.06021066101i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.999 - 0.0380i)T \)
5 \( 1 + (-0.948 - 0.318i)T \)
7 \( 1 + (0.964 - 0.263i)T \)
13 \( 1 + (-0.761 + 0.647i)T \)
17 \( 1 + (0.974 + 0.226i)T \)
19 \( 1 + (0.516 + 0.856i)T \)
23 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (0.123 + 0.992i)T \)
31 \( 1 + (-0.991 + 0.132i)T \)
37 \( 1 + (-0.0285 - 0.999i)T \)
41 \( 1 + (0.640 - 0.768i)T \)
43 \( 1 + (0.580 - 0.814i)T \)
47 \( 1 + (-0.0665 - 0.997i)T \)
53 \( 1 + (-0.362 - 0.931i)T \)
59 \( 1 + (0.345 - 0.938i)T \)
61 \( 1 + (-0.532 + 0.846i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (0.774 + 0.633i)T \)
79 \( 1 + (0.861 + 0.508i)T \)
83 \( 1 + (0.548 + 0.836i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.948 + 0.318i)T \)
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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

−21.563637666314977128357119575971, −20.64836280501020658010968842153, −20.12856915428637509141119887608, −19.27967353110296399809907999052, −18.44758265687500320622289670562, −17.46057149125493237247464270709, −16.54605074028811909235116614688, −15.73771023515744178746958335719, −14.940840019876489300748795149758, −14.605479830641757061791133956587, −13.71249397558852604720592440715, −12.5569414394280290504937528050, −12.090016987456550813096809332, −11.25134238851202274050232734914, −10.732678187498030667733511292121, −9.5526981616842411630376469496, −8.04431612338813848260205650169, −7.77001414217768227573803980953, −6.84454651856414908422805084711, −5.74377121703068055241038932204, −4.853889636783866411130901050414, −4.29702999873840086853987974458, −3.08213554391821485768748570669, −2.512739107931573740671126324674, −1.050394152611019970262154957625, 1.203750977873279766756286956938, 2.11421901270433532742883897447, 3.5682825144827311031460465869, 3.94559272672319036605755229347, 5.124609857191703216601979153450, 5.46479784339245458355764408413, 7.04201015267841258768305084286, 7.496407403806305859434916555454, 8.27085789088998959241759173170, 9.51418600300359704240849644890, 10.67930364328261749241117876714, 11.302792455592290248516326025998, 12.18353153241762322831165437785, 12.45530889646327016262235600937, 13.75057104176794451729547537715, 14.46949424514474605532529532074, 14.893106641018071975145308332359, 15.95959457113193799388702243017, 16.525009266295835117096597494687, 17.30317672166191088828070502870, 18.510916679674051269537013745854, 19.41032383083528110695526193034, 19.991211713728164618455937074597, 20.82453697102398150981262276389, 21.34647159093628141025472286265

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