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Properties

Label 1-133-133.62-r1-0-0
Degree $1$
Conductor $133$
Sign $-0.305 - 0.952i$
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  + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.939 − 0.342i)6-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.766 − 0.642i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (0.939 − 0.342i)13-s + (0.173 − 0.984i)15-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + 18-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.939 − 0.342i)6-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.766 − 0.642i)10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (0.939 − 0.342i)13-s + (0.173 − 0.984i)15-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ -0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.052529773 - 2.813019455i\)
\(L(\frac12)\) \(\approx\) \(2.052529773 - 2.813019455i\)
\(L(1)\) \(\approx\) \(1.754307678 - 1.134240393i\)
\(L(1)\) \(\approx\) \(1.754307678 - 1.134240393i\)

Euler product

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

−29.02550908747029692044349577005, −27.18410537505975913716297121090, −26.28655012139206024811256619921, −25.65027677627902574179221665633, −24.8385456424169106731722975751, −23.54348080128764547089382439286, −22.98170923702373068463826873096, −21.671454736141589160194233279310, −20.79128952038048489923807020094, −19.693048436747817765379163767573, −18.383413443849533609196320728570, −17.659171267363978559806699978430, −15.736624797000087124965734525913, −15.357115070420711804335591639076, −14.128867027522916045814372475, −13.54680009866454106597968017418, −12.3383960963824873476760600507, −11.06438980979639245117395843292, −9.43441961320949033599330984001, −8.074594257025538994873586605059, −7.19083385580164522691068778186, −6.31137136018499227974859891899, −4.48381941620394831551183602805, −3.302930732394587956841442665633, −2.24637700256513339442010762903, 1.02239918999271247624188261185, 2.57604114162626871844547717988, 3.832608706468378034540666889614, 4.78117121426434769324681329709, 6.1468152304444000768658121077, 8.152157646619107974138329172177, 8.97207077818609683938324681363, 10.28597934326759304591297592733, 11.32363381385143927261952187298, 12.9326177461546948048312001685, 13.27749513702194469287124092380, 14.51251421056240918227964880583, 15.634183329540732950922636720097, 16.341024027070239039304876169, 18.30530768121975799621298997136, 19.42673268025644707232276819032, 20.17856141420813101992822568578, 21.02670147651540026110752821489, 21.65661652168235781142390603408, 23.05352815990100000310272586045, 24.19875007347812737869472156241, 24.74946576196209656365149847689, 26.08023416596582459469203218427, 27.27936913985296330303165988817, 28.18549611985563238165776552810

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