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Properties

Label 1-133-133.69-r0-0-0
Degree $1$
Conductor $133$
Sign $-0.0977 - 0.995i$
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  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + 11-s + 12-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + 11-s + 12-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.0977 - 0.995i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ -0.0977 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7702517748 - 0.8496007027i\)
\(L(\frac12)\) \(\approx\) \(0.7702517748 - 0.8496007027i\)
\(L(1)\) \(\approx\) \(0.9900990506 - 0.5583941372i\)
\(L(1)\) \(\approx\) \(0.9900990506 - 0.5583941372i\)

Euler product

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

−29.18035169496956610480641354014, −27.82060246806366464833803331839, −26.5836822575357451321328536137, −25.64755111799824479482593247631, −24.84006169012581285104358771383, −23.918948094264342861118972759029, −22.99188962375765912315330739563, −22.13736747029761034196785477240, −21.409386837270804456900006397706, −19.42063194848422660475205203485, −18.58553551196302344889316422131, −17.2868755813870329501713545880, −17.07265580125286838481850066088, −15.43973259423463330175062655512, −14.17903273793567544108612385506, −13.75784399786680694994112623180, −12.31934401239585820625808943015, −11.53183565145392270396513742669, −9.87084018350060669385403932975, −8.33872677984762376193890129622, −7.0411486842574567757584094729, −6.462228372672516107937559638109, −5.404062871843626007146098235703, −3.73894782180188483967873331797, −2.039594311284700866984196392618, 1.034829419770963066005101409010, 2.95513442887018324488165742536, 4.38357994174166881425518825067, 5.18001079919175838623389714945, 6.266560514119405304364294991126, 8.69220009497401116065253492946, 9.66696210101834828531994552600, 10.42387834802669532515818207339, 11.83571148321414984678044486381, 12.42626300508625871151676089628, 13.84787559626758372145739017820, 14.80585366705969630155714470588, 16.08628839092649842893819806712, 17.16622471806108860189763391030, 18.111290254937319890062212218955, 19.76731080471400668157929615815, 20.45959654964663228366068527322, 21.329552010855352777371885817508, 22.24254894059549593527039216084, 22.941277145414557968453672466411, 24.22264799055344565736544015857, 25.18377443159592776530195350803, 26.8658319863526877600559071345, 27.68864787180541299602202666445, 28.3129001310499678162057441918

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