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Properties

Label 1-33e2-1089.119-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.816 - 0.576i$
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  + (−0.449 − 0.893i)2-s + (−0.595 + 0.803i)4-s + (−0.999 − 0.0380i)5-s + (0.123 + 0.992i)7-s + (0.985 + 0.170i)8-s + (0.415 + 0.909i)10-s + (0.00951 − 0.999i)13-s + (0.830 − 0.556i)14-s + (−0.290 − 0.956i)16-s + (−0.941 + 0.336i)17-s + (−0.0285 + 0.999i)19-s + (0.625 − 0.780i)20-s + (−0.981 + 0.189i)23-s + (0.997 + 0.0760i)25-s + (−0.897 + 0.441i)26-s + ⋯
L(s)  = 1  + (−0.449 − 0.893i)2-s + (−0.595 + 0.803i)4-s + (−0.999 − 0.0380i)5-s + (0.123 + 0.992i)7-s + (0.985 + 0.170i)8-s + (0.415 + 0.909i)10-s + (0.00951 − 0.999i)13-s + (0.830 − 0.556i)14-s + (−0.290 − 0.956i)16-s + (−0.941 + 0.336i)17-s + (−0.0285 + 0.999i)19-s + (0.625 − 0.780i)20-s + (−0.981 + 0.189i)23-s + (0.997 + 0.0760i)25-s + (−0.897 + 0.441i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.816 - 0.576i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.816 - 0.576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06413220770 - 0.2020287735i\)
\(L(\frac12)\) \(\approx\) \(0.06413220770 - 0.2020287735i\)
\(L(1)\) \(\approx\) \(0.5658353227 - 0.1125078238i\)
\(L(1)\) \(\approx\) \(0.5658353227 - 0.1125078238i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.449 - 0.893i)T \)
5 \( 1 + (-0.999 - 0.0380i)T \)
7 \( 1 + (0.123 + 0.992i)T \)
13 \( 1 + (0.00951 - 0.999i)T \)
17 \( 1 + (-0.941 + 0.336i)T \)
19 \( 1 + (-0.0285 + 0.999i)T \)
23 \( 1 + (-0.981 + 0.189i)T \)
29 \( 1 + (0.432 - 0.901i)T \)
31 \( 1 + (0.749 + 0.662i)T \)
37 \( 1 + (-0.736 + 0.676i)T \)
41 \( 1 + (-0.964 - 0.263i)T \)
43 \( 1 + (-0.786 + 0.618i)T \)
47 \( 1 + (0.935 + 0.353i)T \)
53 \( 1 + (-0.974 + 0.226i)T \)
59 \( 1 + (0.710 + 0.703i)T \)
61 \( 1 + (0.548 + 0.836i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (-0.610 + 0.791i)T \)
73 \( 1 + (0.516 - 0.856i)T \)
79 \( 1 + (-0.969 - 0.244i)T \)
83 \( 1 + (0.905 - 0.424i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.999 - 0.0380i)T \)
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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

−21.80046444196552821170007674407, −20.36021027365966073050340173845, −19.93106004745910230117739721951, −19.16161687464573973835246567921, −18.415698449128370818847224165920, −17.51423814293004994943178387163, −16.86185277115337228765637437483, −15.9907333305270707271455310799, −15.60674587024435195414522064211, −14.56766059651580713277407169082, −13.90186697633154030264554755158, −13.191576052637496393267598268922, −11.91737318606514368095747170019, −11.0975662716464747082432631275, −10.39951121142000675580341638718, −9.35183386962560545983521460089, −8.55348383960715405573532844593, −7.811980407262455576039128525, −6.82897212297543919694955926352, −6.670323658682507805419979880002, −5.01339261487921695314933302738, −4.42685613327131795552176557625, −3.62597646881710094105847946256, −1.980709987380845325200022546286, −0.66705538708264832129137251781, 0.08162069523916534570725961432, 1.37082272418996279468268625338, 2.469266568733590339482484756934, 3.3068163834042976208396900461, 4.19287322442722383906969711701, 5.09874979929657448744972415971, 6.257790985179929889064204861642, 7.54504792158899155724857891400, 8.36476282680247881763549778999, 8.637904883938855085751522229119, 9.93567458190409500777256772508, 10.54516772097928364206650860476, 11.620816945614127369035697492708, 12.007161952450666961459333769325, 12.72317345569408590753815077879, 13.624349054552165003464624105160, 14.771415961140410429191891126234, 15.58116556451601740759174915727, 16.16932123726582838150535591445, 17.36977551368776371225141054971, 17.93535750479255265151826136424, 18.84273332170066637045878688894, 19.284105099394210788222635739674, 20.1797463075846542253452932991, 20.68779818411726272294738737992

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