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Properties

Label 1-33-33.2-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: $\chi_{33} (2, \cdot )$
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

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

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