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Properties

Label 1-33e2-1089.412-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  + (0.272 − 0.962i)2-s + (−0.851 − 0.524i)4-s + (0.00951 − 0.999i)5-s + (−0.935 − 0.353i)7-s + (−0.736 + 0.676i)8-s + (−0.959 − 0.281i)10-s + (0.380 + 0.924i)13-s + (−0.595 + 0.803i)14-s + (0.449 + 0.893i)16-s + (0.0855 + 0.996i)17-s + (−0.921 − 0.389i)19-s + (−0.532 + 0.846i)20-s + (0.0475 + 0.998i)23-s + (−0.999 − 0.0190i)25-s + (0.993 − 0.113i)26-s + ⋯
L(s)  = 1  + (0.272 − 0.962i)2-s + (−0.851 − 0.524i)4-s + (0.00951 − 0.999i)5-s + (−0.935 − 0.353i)7-s + (−0.736 + 0.676i)8-s + (−0.959 − 0.281i)10-s + (0.380 + 0.924i)13-s + (−0.595 + 0.803i)14-s + (0.449 + 0.893i)16-s + (0.0855 + 0.996i)17-s + (−0.921 − 0.389i)19-s + (−0.532 + 0.846i)20-s + (0.0475 + 0.998i)23-s + (−0.999 − 0.0190i)25-s + (0.993 − 0.113i)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: $\chi_{1089} (412, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.999 + 0.00576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7630322037 + 0.002201233255i\)
\(L(\frac12)\) \(\approx\) \(0.7630322037 + 0.002201233255i\)
\(L(1)\) \(\approx\) \(0.7464973897 - 0.4258725086i\)
\(L(1)\) \(\approx\) \(0.7464973897 - 0.4258725086i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.272 - 0.962i)T \)
5 \( 1 + (0.00951 - 0.999i)T \)
7 \( 1 + (-0.935 - 0.353i)T \)
13 \( 1 + (0.380 + 0.924i)T \)
17 \( 1 + (0.0855 + 0.996i)T \)
19 \( 1 + (-0.921 - 0.389i)T \)
23 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (0.483 - 0.875i)T \)
31 \( 1 + (-0.179 + 0.983i)T \)
37 \( 1 + (-0.564 + 0.825i)T \)
41 \( 1 + (-0.0665 + 0.997i)T \)
43 \( 1 + (0.580 - 0.814i)T \)
47 \( 1 + (0.640 + 0.768i)T \)
53 \( 1 + (-0.998 + 0.0570i)T \)
59 \( 1 + (-0.830 + 0.556i)T \)
61 \( 1 + (-0.969 - 0.244i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (0.974 - 0.226i)T \)
73 \( 1 + (-0.254 - 0.967i)T \)
79 \( 1 + (0.749 - 0.662i)T \)
83 \( 1 + (-0.625 + 0.780i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (0.00951 + 0.999i)T \)
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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

−21.75449050198100430963406097260, −20.85483586242923596689139399051, −19.75222021870506862277785684768, −18.64229400761457614741958982594, −18.49686205199912583387288787374, −17.50093133142083974798729854394, −16.63448153796227708995383914073, −15.77566730851105791456612533540, −15.334546131752409733926384744393, −14.42124603895628683220428277435, −13.82890076055582683036691696983, −12.797054527271284620947954453087, −12.33478851946411858467991029283, −11.00683141533318261958747405398, −10.18500129615934808841230700981, −9.33370081897082028242766918540, −8.445960766785482471005598863102, −7.509317425714647937991652159905, −6.72550653581291434700268234771, −6.10476477299513542642549062646, −5.32253199874057556900629909635, −4.06072141800502943407306768671, −3.23198418406687552887878744400, −2.478258212573814725762499033570, −0.32074324444460006360275868682, 1.11591889352045262604147117460, 1.94779679865590397097604848391, 3.220455116716460478948558578134, 4.06365975368929194121330588294, 4.69192246305632435107923846850, 5.84645698334359227098133808179, 6.55628706688413636241648484665, 8.01475171185124716303287077692, 8.91234179919657352392003144344, 9.45530404265025641116640031991, 10.3299992790279867994900322016, 11.14770747547848318057341031919, 12.16022230645110579296164123995, 12.64917332004469568859811409552, 13.48255454325922640388585482973, 13.9234385157489236954079671618, 15.20779819927562481477698975006, 15.94177478344021266432742219721, 17.020534305435805848221777693685, 17.38980366251651364825941785726, 18.6890068733168121399352552069, 19.42665729878071789877026465903, 19.74531482884582937996128453944, 20.7237695933119524401814266267, 21.39674986622189931784310716006

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