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Properties

Label 1-33e2-1089.520-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.258 + 0.965i$
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  + (−0.217 + 0.976i)2-s + (−0.905 − 0.424i)4-s + (0.879 + 0.475i)5-s + (0.999 − 0.0380i)7-s + (0.610 − 0.791i)8-s + (−0.654 + 0.755i)10-s + (0.123 − 0.992i)13-s + (−0.179 + 0.983i)14-s + (0.640 + 0.768i)16-s + (−0.254 + 0.967i)17-s + (−0.362 + 0.931i)19-s + (−0.595 − 0.803i)20-s + (−0.786 − 0.618i)23-s + (0.548 + 0.836i)25-s + (0.941 + 0.336i)26-s + ⋯
L(s)  = 1  + (−0.217 + 0.976i)2-s + (−0.905 − 0.424i)4-s + (0.879 + 0.475i)5-s + (0.999 − 0.0380i)7-s + (0.610 − 0.791i)8-s + (−0.654 + 0.755i)10-s + (0.123 − 0.992i)13-s + (−0.179 + 0.983i)14-s + (0.640 + 0.768i)16-s + (−0.254 + 0.967i)17-s + (−0.362 + 0.931i)19-s + (−0.595 − 0.803i)20-s + (−0.786 − 0.618i)23-s + (0.548 + 0.836i)25-s + (0.941 + 0.336i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (520, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.258 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9643091393 + 1.256524489i\)
\(L(\frac12)\) \(\approx\) \(0.9643091393 + 1.256524489i\)
\(L(1)\) \(\approx\) \(0.9602125262 + 0.6070071682i\)
\(L(1)\) \(\approx\) \(0.9602125262 + 0.6070071682i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.217 + 0.976i)T \)
5 \( 1 + (0.879 + 0.475i)T \)
7 \( 1 + (0.999 - 0.0380i)T \)
13 \( 1 + (0.123 - 0.992i)T \)
17 \( 1 + (-0.254 + 0.967i)T \)
19 \( 1 + (-0.362 + 0.931i)T \)
23 \( 1 + (-0.786 - 0.618i)T \)
29 \( 1 + (0.449 + 0.893i)T \)
31 \( 1 + (-0.999 + 0.0190i)T \)
37 \( 1 + (0.974 - 0.226i)T \)
41 \( 1 + (-0.948 - 0.318i)T \)
43 \( 1 + (0.723 + 0.690i)T \)
47 \( 1 + (0.00951 + 0.999i)T \)
53 \( 1 + (-0.985 - 0.170i)T \)
59 \( 1 + (0.749 + 0.662i)T \)
61 \( 1 + (0.953 + 0.299i)T \)
67 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (0.774 + 0.633i)T \)
73 \( 1 + (0.696 - 0.717i)T \)
79 \( 1 + (0.997 + 0.0760i)T \)
83 \( 1 + (-0.830 - 0.556i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.879 - 0.475i)T \)
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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

−21.32225840019517667958230754548, −20.39245757759722042451029251392, −19.95372694226096080367394955014, −18.763695336077175543573322347781, −18.18194396666592971455180346703, −17.456428043111037016165791085192, −16.92025827471321396500645099150, −15.895021255334039481865068215423, −14.62606154733198952944460555174, −13.81664089174503864550584301093, −13.46350837039921719756965968144, −12.39209764122373869684590595043, −11.55042291792068510290014275444, −11.07055355028831939736015691460, −9.94531487568774930697249974872, −9.30764137648767984230806770593, −8.63969612137403789173052130623, −7.74750604760359471940938197079, −6.55965683800269440911424460698, −5.27930253740870374619816582786, −4.742987834902086041838158489482, −3.8259810294370023344342073241, −2.335584387011801480149759768764, −1.95791434950994310505100786185, −0.801081851455791480888104925254, 1.21275012277288311287719393377, 2.15096164206469338505035544910, 3.6052661287491479734872666503, 4.63013595541540258942229942691, 5.61678881237303205476429580947, 6.081913876734620488721850163604, 7.090920749694417209552034875585, 8.05282194711541350046266480371, 8.53837746538121966476425371705, 9.658730769651097981339828918014, 10.463675655172575653870705228112, 10.95624888204322694403483752243, 12.525285787144462789020798617646, 13.131660736823317058498676998495, 14.30209416622070138543557776752, 14.47963797213354005754325226746, 15.2770485897660634864071037638, 16.313821167924644704324988524382, 17.099804717952997828670176245438, 17.84439244752180495464269335616, 18.14496513906777433914848928204, 19.03666256406635235238108897732, 20.12504263072985092661685208124, 20.981133084808812116545555270177, 21.910083384572416677430712398

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