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Properties

Label 1-133-133.80-r1-0-0
Degree $1$
Conductor $133$
Sign $0.423 - 0.906i$
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  + (−0.939 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.423 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.423 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.423 - 0.906i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ 0.423 - 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5973598949 - 0.3803457188i\)
\(L(\frac12)\) \(\approx\) \(0.5973598949 - 0.3803457188i\)
\(L(1)\) \(\approx\) \(0.5485329969 - 0.1126674800i\)
\(L(1)\) \(\approx\) \(0.5485329969 - 0.1126674800i\)

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

−28.17333537352820390598351614262, −27.61496568308484337115483794947, −26.86428321949398795929057177809, −25.97892670110916633662270576435, −24.748086237090448023941408351355, −23.3363321072658047664905636329, −22.461139231364229492239755000021, −21.5058138768927700600782182059, −20.34439080808948053366977309798, −19.44508197288053298649592492798, −18.20250975699113268380225699455, −17.59053717871978084649171989753, −16.11320041025502417375223615970, −15.828090908242604225258266544332, −14.38437227691017535470026644426, −12.35697761518465580014906640849, −11.49924901608056324984076733347, −10.81685912421903701396974133371, −9.71425104781970215903182911662, −8.559591394664292252513924131529, −7.135145526602149433064290470812, −6.16578289488748564554625157885, −4.18395856216475185332938428382, −3.13329953415290371101059146074, −0.94463837498746044952543641814, 0.59890034394377832618898501492, 1.74365101067064620958468177312, 4.192237048494855680665094669914, 5.84726859976395934062029104501, 6.72871034358423730358133644063, 7.99834202995094472704288170506, 8.792259140604592425033934740548, 10.3607954499491032378482471972, 11.53477250297499998555735156866, 12.140305985730917763470073944087, 13.65380818127138535048981259127, 15.21341092978315259764057888428, 16.282560609745391606319356706423, 16.962014055433060896498773523183, 17.922173992832483655095411174, 19.087802656781047109709871020658, 19.64368665879375259489719730376, 20.88719369742114445176906853523, 22.48486501279172226123120465120, 23.6092679079443146789122340226, 24.201817513548482988022207516749, 25.074147072966330851105257015326, 26.25933176968537400569710945178, 27.54615391804701045863281733960, 28.08170199327434426964157151337

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