Properties

Label 1-133-133.17-r1-0-0
Degree $1$
Conductor $133$
Sign $-0.832 + 0.554i$
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 + (−0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.173 + 0.984i)6-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.766 − 0.642i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 + 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.173 + 0.984i)6-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.766 − 0.642i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 + 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ -0.832 + 0.554i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1160662268 - 0.3839051652i\)
\(L(\frac12)\) \(\approx\) \(-0.1160662268 - 0.3839051652i\)
\(L(1)\) \(\approx\) \(0.7991376746 - 0.3857221154i\)
\(L(1)\) \(\approx\) \(0.7991376746 - 0.3857221154i\)

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 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−29.44650948552911720750836555085, −27.9701051843581892639212334823, −26.7204638794773936233004500602, −25.870472912885255400231664485422, −24.68606757588535303792996169670, −23.88869484123814533612760023301, −23.01829086845868131832444350555, −22.22338691831801828181053861636, −21.49402548866378318928403404470, −19.80919124419820250610633754448, −18.51227288301828826964491051873, −17.755024627216160433518554387134, −16.644299435739187130254604620613, −15.60196843691561156641983063365, −14.53846031407883647173186983991, −13.415846826405176984839462364966, −12.55205934707873678260086177990, −11.33641250060094452486051868163, −10.52482945341818127999897460721, −8.18646387183121969569979029424, −7.39658512771873769448781248262, −6.18922394014544260536510712691, −5.52142377209636138602054478072, −3.79903104689256804590742088464, −2.41713382455092040900885285255, 0.12692743970062614654138039927, 1.84379502018486748105635780480, 3.82968022963815236118652541769, 4.76449726738890858859840249336, 5.54003048415216480868350935719, 7.0586586960674846562468218379, 9.17694036735985577062784878812, 9.93294722644051109267108181411, 11.28323217430796255464311163725, 12.06731593242770771439233677099, 12.93329675994149038508327997586, 14.27909423531060880421982915933, 15.58215231944029875630731904490, 16.23324145926843481017217707711, 17.52052961317293413878609536186, 18.76668882493619095031349129475, 20.27937922734242081044221240380, 20.69628202743820139360722662197, 21.78397619748299701966130657148, 22.65729421645679157525237618636, 23.71249377157307325886613381235, 24.21837652541214020403949113628, 25.78094532281319408282184242302, 27.32421930391283771342410998316, 27.98022505270289503682475880258

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