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Properties

Label 1-133-133.44-r0-0-0
Degree $1$
Conductor $133$
Sign $0.605 + 0.796i$
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 + (−0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (0.766 − 0.642i)6-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.939 − 0.342i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.766 − 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (−0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (0.766 − 0.642i)6-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.939 − 0.342i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.766 − 0.642i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (−0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.605 + 0.796i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ 0.605 + 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5553527072 + 0.2754556080i\)
\(L(\frac12)\) \(\approx\) \(0.5553527072 + 0.2754556080i\)
\(L(1)\) \(\approx\) \(0.6148402300 + 0.1896694014i\)
\(L(1)\) \(\approx\) \(0.6148402300 + 0.1896694014i\)

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

−28.38813178091580386421615112469, −27.96764880490364836847902165227, −26.592539296947390598339079991212, −25.45898430130980433321871955847, −24.71904786609435989403616562402, −23.58823622639511582953925262829, −22.38949162036372533745742134674, −21.08934326520715600857852382518, −20.6312566015715443864336481852, −19.019362234718526013737186634389, −18.232113747998892837086086698201, −17.36178968796098199595386927156, −16.59177223719822884901245240378, −15.6761247079100464424162531729, −13.64449505116835351996917703842, −12.55027275125511392415923027191, −11.77165090844744734374009149734, −10.46096051276151115133392622120, −9.6871105622998556752364714915, −8.35686010487381045473720147744, −7.051484524140679037606595291203, −5.975604479156728778407697373470, −4.561267416230016471860210082989, −2.30562149354751800509039574023, −1.070945019550151272921466272331, 1.27540182655890781221562441771, 3.190830081174200307568608777165, 5.51422346100219201443527008774, 6.022724437419223527728506903791, 7.28172900025988903084137078846, 8.72008220914687516652948691195, 10.10349580232851871135116053443, 10.613813596426649000958535066306, 11.61425348403844281478763965227, 13.256851355685580028713013044648, 14.69632601713080836079138734985, 15.75651686649639591258475352538, 16.65028243424219407347662776356, 17.683455640643155643294501049178, 18.27615215518031476757810737837, 19.289995548640066063283612635721, 20.95001415318515836729248333276, 21.56783234835760333626926994417, 22.95286987205542391460514721865, 23.76597979981777139142686286955, 24.9940083163881225200305126165, 25.980823357474491568384126786111, 26.79053672822149931250632025388, 27.79016544726574096741980183695, 28.59687112137865720305909898766

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