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Properties

Label 1-133-133.81-r0-0-0
Degree $1$
Conductor $133$
Sign $-0.781 - 0.623i$
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  + (0.173 + 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (−0.939 + 0.342i)6-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.173 − 0.984i)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 + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (−0.939 + 0.342i)6-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.173 − 0.984i)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 + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.781 - 0.623i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ -0.781 - 0.623i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1829937174 + 0.5225208777i\)
\(L(\frac12)\) \(\approx\) \(-0.1829937174 + 0.5225208777i\)
\(L(1)\) \(\approx\) \(0.4136102219 + 0.6020695328i\)
\(L(1)\) \(\approx\) \(0.4136102219 + 0.6020695328i\)

Euler product

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

−28.34404828509934226688713064709, −26.92751141414396379531987323211, −26.48077843319751079598098284292, −24.70136806061063906436902847766, −23.86994341627752972977092524851, −22.9945517244920426493908732845, −22.10858278819706038730846742228, −20.75929220735514009191635422381, −19.62609965671003787978067669885, −19.19993590271348410270365073418, −18.24968530305435020265190857117, −17.17855689044275911725276428783, −15.38819598645261968888960312550, −14.31434474826190224477440027127, −13.26310852764606230893076779978, −12.326031871504191886163299035339, −11.44408160398749784570817290663, −10.47241429461024773777323451621, −8.74948025925440360963989909433, −7.92086040543439648375947091638, −6.49482221439971921921323465552, −4.889380171055606899675926043792, −3.32834035194494571532774541991, −2.3726680741870738898363144999, −0.45388482263377191762801906062, 3.11064742486757508218272089902, 4.59518265257015867335897122457, 4.926845388954893857759873511591, 6.84180584007075754820448709933, 7.957744008283952970988356305178, 8.991907742838720340066463819510, 10.011713368065587272267390148024, 11.57311278998289278524606729278, 12.77497977041018128725692967460, 14.136200343697956503368886214665, 15.27500432763044718997591035155, 15.63775063409641838503806783650, 16.75291398214676851146869220563, 17.64301632389907486500205570336, 19.195121329889370473239665647492, 20.22635947520377217888675090257, 21.374218238870788608029331394517, 22.46273337615162096881818137131, 23.20016063200933885696220559877, 24.24042608780345196876969906269, 25.31164708244944279708643676004, 26.37421917219053476067856967889, 27.056812976857215943560456625449, 27.81434964885677552398740681346, 28.87565837011436417586973745942

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