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Properties

Label 1-33-33.17-r0-0-0
Degree $1$
Conductor $33$
Sign $0.569 + 0.821i$
Analytic cond. $0.153251$
Root an. cond. $0.153251$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s − 10-s + (0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.809 − 0.587i)20-s − 23-s + (0.309 + 0.951i)25-s + (−0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s − 10-s + (0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.809 − 0.587i)20-s − 23-s + (0.309 + 0.951i)25-s + (−0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(0.153251\)
Root analytic conductor: \(0.153251\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 33,\ (0:\ ),\ 0.569 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5220111441 + 0.2732471404i\)
\(L(\frac12)\) \(\approx\) \(0.5220111441 + 0.2732471404i\)
\(L(1)\) \(\approx\) \(0.7015318160 + 0.2509844870i\)
\(L(1)\) \(\approx\) \(0.7015318160 + 0.2509844870i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 - T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−36.19795842229242896817721890143, −35.52969320214527245299562512740, −33.781215669954199845463891946949, −32.731490138180666444257290185120, −31.05328893856030169218815241303, −29.70060153288278081260466466165, −28.9178388531692312669786705775, −27.75681103569195776545650782240, −26.319916036305950845800501163637, −25.47418249492842928578833936684, −23.8946097295547947659143958539, −22.05245927027376060949814619880, −20.816291794910386017249347092890, −19.9121274558279126157875437860, −18.37063877133369887682667207669, −17.11911729905585608593536605753, −16.230056945116934991768369384311, −13.77216186631519702968103691531, −12.62887137314387567892480254050, −10.8556483604253559786035201989, −9.69028792982630779375740460053, −8.33919258951056183928666948583, −6.49004810281006868829295876850, −3.97404342706302846846083356798, −1.67495966336020647089026650469, 2.36743855028935513143417571498, 5.59158315813362444915143074935, 6.71840053017264650528317143821, 8.61053599690987502322113858622, 9.81896909829894142043448044882, 11.2268009844645426844532498981, 13.434002833179710045198777613, 14.95328445254739813837861523824, 16.03861466813170788450388596824, 17.76803907073547066181837636087, 18.387038424790176875065120071577, 19.86687537893560831286067351158, 21.59327210065935975073621041227, 22.93660919224192186323165433081, 24.63646418695947586655673565014, 25.510945751166654174480676250211, 26.47654583375913635998754805683, 28.006983239299727084308048087360, 28.90876645372612595978470592967, 30.26686849931785263364621699571, 32.07327072383274115934409515863, 33.20793565067470220281345195477, 34.281923732387196123468822972612, 35.233277247426692398009758992080, 36.541569137908278893910308707948

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