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Properties

Label 1-33e2-1089.191-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.514 - 0.857i$
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  + (0.851 − 0.524i)2-s + (0.449 − 0.893i)4-s + (0.999 − 0.0190i)5-s + (0.749 − 0.662i)7-s + (−0.0855 − 0.996i)8-s + (0.841 − 0.540i)10-s + (−0.710 − 0.703i)13-s + (0.290 − 0.956i)14-s + (−0.595 − 0.803i)16-s + (0.985 + 0.170i)17-s + (0.696 − 0.717i)19-s + (0.432 − 0.901i)20-s + (0.995 + 0.0950i)23-s + (0.999 − 0.0380i)25-s + (−0.974 − 0.226i)26-s + ⋯
L(s)  = 1  + (0.851 − 0.524i)2-s + (0.449 − 0.893i)4-s + (0.999 − 0.0190i)5-s + (0.749 − 0.662i)7-s + (−0.0855 − 0.996i)8-s + (0.841 − 0.540i)10-s + (−0.710 − 0.703i)13-s + (0.290 − 0.956i)14-s + (−0.595 − 0.803i)16-s + (0.985 + 0.170i)17-s + (0.696 − 0.717i)19-s + (0.432 − 0.901i)20-s + (0.995 + 0.0950i)23-s + (0.999 − 0.0380i)25-s + (−0.974 − 0.226i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.514 - 0.857i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.514 - 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.687857267 - 4.748374208i\)
\(L(\frac12)\) \(\approx\) \(2.687857267 - 4.748374208i\)
\(L(1)\) \(\approx\) \(1.961131290 - 1.288709984i\)
\(L(1)\) \(\approx\) \(1.961131290 - 1.288709984i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.851 - 0.524i)T \)
5 \( 1 + (0.999 - 0.0190i)T \)
7 \( 1 + (0.749 - 0.662i)T \)
13 \( 1 + (-0.710 - 0.703i)T \)
17 \( 1 + (0.985 + 0.170i)T \)
19 \( 1 + (0.696 - 0.717i)T \)
23 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (0.532 - 0.846i)T \)
31 \( 1 + (-0.935 + 0.353i)T \)
37 \( 1 + (-0.362 + 0.931i)T \)
41 \( 1 + (0.991 - 0.132i)T \)
43 \( 1 + (-0.327 + 0.945i)T \)
47 \( 1 + (0.179 + 0.983i)T \)
53 \( 1 + (-0.993 - 0.113i)T \)
59 \( 1 + (-0.380 - 0.924i)T \)
61 \( 1 + (0.879 - 0.475i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (-0.897 - 0.441i)T \)
73 \( 1 + (-0.870 - 0.491i)T \)
79 \( 1 + (0.123 + 0.992i)T \)
83 \( 1 + (0.217 - 0.976i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (-0.999 - 0.0190i)T \)
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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

−21.669890400314181690612166698979, −20.94177641520192635422818833864, −20.412818815450477824224710354344, −19.04541635616904557009037134777, −18.214896625616016557312967294969, −17.49258796353025281217713256139, −16.72155981930024023312184950499, −16.11355862382567501709623577104, −14.89738977431321246300303340617, −14.43625737356051105111681194465, −13.935637086613118227742137635218, −12.84339487066529671887200090729, −12.218108601642232243781292286, −11.450290436609271447274406407320, −10.442296759570132895934460590469, −9.334089680020141506798163357034, −8.64198157665734189854178365662, −7.51603198528711189330606545020, −6.88224256221580962719652458185, −5.5628293746912761156741828298, −5.4703237814369588829135246310, −4.42688805354045026206193598536, −3.19107381131751247978561426741, −2.31012720879700623735926856365, −1.43791288890434387640227426941, 0.796281879341827927511863882221, 1.516336495737742343186584171496, 2.63215258482745258099337946970, 3.38013602693732688256779510257, 4.76158372609564255897382616479, 5.119020087873861268790857791175, 6.07193674089599876330621597889, 7.05576091414904072571736795585, 7.90120358180662664494673832564, 9.32814650235696160304516314169, 9.96377056477576503736163630121, 10.73026941947418459322521396376, 11.419899237021845997685359294947, 12.48440075288686917289029918003, 13.106252175100352899721239711214, 13.94786215981929151541924336699, 14.45146397162731877930666860881, 15.17974318646271087376461969673, 16.27068426383600841363658263021, 17.256542279229845160772134971874, 17.77029093576669348495199587423, 18.79718114311096968145494584165, 19.65442650958318880054146919462, 20.513848332973444542190214325387, 20.94889928475874128438615327080

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