↑

Properties

Label 1-33e2-1089.302-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

Related objects

Downloads

Learn more

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: $\chi_{1089} (302, \cdot )$
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 \( 1 + (0.999 + 0.0380i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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