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Properties

Label 1-133-133.82-r1-0-0
Degree $1$
Conductor $133$
Sign $-0.978 - 0.205i$
Analytic cond. $14.2928$
Root an. cond. $14.2928$
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  + (−0.939 + 0.342i)2-s + (0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.766 + 0.642i)6-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (0.939 + 0.342i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (−0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.766 + 0.642i)6-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (0.939 + 0.342i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.766 + 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (−0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.978 - 0.205i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ -0.978 - 0.205i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04382980453 - 0.4216297389i\)
\(L(\frac12)\) \(\approx\) \(0.04382980453 - 0.4216297389i\)
\(L(1)\) \(\approx\) \(0.6411558106 - 0.1569700196i\)
\(L(1)\) \(\approx\) \(0.6411558106 - 0.1569700196i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−28.61170166260997103494014257375, −27.59267118589589598681248895599, −26.78169094301070052077879442749, −26.1753492558387388637611048782, −25.22813652006436110637912826406, −24.17835137564604101899255688415, −22.59355581091162557038017500584, −21.588284354561589135293890150935, −20.3197723632573806773129625475, −19.84974709502560568847360473939, −18.84812385066013514485781305123, −17.95510079698802317215624932906, −16.58948436077633288091213273881, −15.22795413214450813067324548766, −15.04662808717695325466015733525, −13.155453328482348816068517715234, −12.03347239195690623952098451261, −10.60331344267383058614657754348, −10.018349433118266910297089984661, −8.62410347826466828358851709312, −7.760979583960537163782469152760, −6.86964939675613359910902243267, −4.42836531379083298747626725714, −3.11721472130649358135632598759, −2.13205161672140158883528738933, 0.18979522857209789804288369775, 1.77114226094641901412150743744, 3.29531355013923491075424913261, 5.075048021881858321115199862588, 6.89061136657828010997201712243, 7.73890079887266694013758252844, 8.74243736104965033702059752134, 9.436890076984964346320125379033, 11.028078691793739380986234339391, 12.16746492612627686029925726776, 13.527216710445379455129030347986, 14.72918148741363596930833519135, 15.75220370224763215632199388650, 16.50385525458439057509788458350, 17.899712209653584043550032992917, 18.957769135852510454946137401891, 19.64163562647820293692290358889, 20.41035226611596374588995403106, 21.51763060947553110552108239013, 23.54598351835968101853543861885, 24.1897488012504752416953582893, 24.904510581306276771380619530200, 26.06973815142951163177731022324, 26.890142332549039367771339802, 27.52911433201416360322306823889

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