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Properties

Label 1-133-133.4-r0-0-0
Degree $1$
Conductor $133$
Sign $0.273 + 0.961i$
Analytic cond. $0.617649$
Root an. cond. $0.617649$
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 + (0.766 + 0.642i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)6-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.939 − 0.342i)10-s + 11-s + 12-s + (−0.939 − 0.342i)13-s + (0.173 + 0.984i)15-s + (0.173 − 0.984i)16-s + (0.173 − 0.984i)17-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)6-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.939 − 0.342i)10-s + 11-s + 12-s + (−0.939 − 0.342i)13-s + (0.173 + 0.984i)15-s + (0.173 − 0.984i)16-s + (0.173 − 0.984i)17-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.273 + 0.961i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ 0.273 + 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8205222651 + 0.6198346781i\)
\(L(\frac12)\) \(\approx\) \(0.8205222651 + 0.6198346781i\)
\(L(1)\) \(\approx\) \(0.9021207410 + 0.4152816846i\)
\(L(1)\) \(\approx\) \(0.9021207410 + 0.4152816846i\)

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 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
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

−28.46106369544857836086986395229, −27.45578276362062135188104443555, −26.27641614177506358250773246704, −25.52841634456728309855775708377, −24.6704136904014412094059552634, −24.01400633384608158986988590125, −21.94260892135656858452733630315, −21.15778798357647972659066057230, −19.93008303718881887777381133154, −19.56486002721229033918200255087, −18.27226533501060062947820329327, −17.3725473951741466264614394955, −16.57469595600760062509459622912, −14.98906946775267019162552910919, −13.864064724617035776666565669399, −12.60619315242167321336090282105, −11.93064679215548093465994154861, −10.14469442478746623171225684937, −9.25243088112665276372025171751, −8.46202636220850407497913802129, −7.22946244481229526688672792803, −6.098141166582757119711994362310, −3.9045615756188256951840629568, −2.28993969644975572284769234884, −1.36806324122690147389598244291, 1.94199465046455988695179325012, 3.09168087055960253912693625468, 5.04014895437370770224828990012, 6.50169230040709327953107968669, 7.599629459576148047347221117697, 8.90778242605988387808296353561, 9.78785200481011047692796955737, 10.446928250635989921708941249482, 11.85874897004937048303803817710, 13.90524389520894584572944169667, 14.50947196525144379378087435104, 15.506881593355343103859068007779, 16.66540861851145176498838909198, 17.59381731333170384811050354780, 18.716939491456518917254608506477, 19.69073219012111321261301402193, 20.52847896740248572178050116072, 21.71689795914119077206598528826, 22.6104536181744314624340859512, 24.4203402701701994812113236613, 25.19379886603568810804593780630, 25.81932801629815118704170663248, 26.99148699602154100565817988542, 27.30771403612963821018679998558, 28.69273911797118723574336605867

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