Properties

Label 1-133-133.33-r0-0-0
Degree $1$
Conductor $133$
Sign $0.941 + 0.336i$
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 + (0.766 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ 0.941 + 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.024054025 + 0.3502797162i\)
\(L(\frac12)\) \(\approx\) \(2.024054025 + 0.3502797162i\)
\(L(1)\) \(\approx\) \(1.876570655 + 0.2307474342i\)
\(L(1)\) \(\approx\) \(1.876570655 + 0.2307474342i\)

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

−28.43235810541884132058436277839, −27.69930219924855478861541416941, −26.71935554470435997823612452665, −25.28770952793491214962069079456, −24.52495235985299693229786573816, −23.59835842142833452267122356455, −22.23757480447475009925429540622, −21.70264275336905073300448166349, −20.296711452042692759166171975844, −19.94281669011683125922293231190, −19.07971898017232362792870417533, −16.92008914855814480038081652923, −15.98347433039361742262102128963, −14.96508752362446036126468913867, −14.33241487310772603735064562440, −12.95906130457096726985071752760, −12.09057350325030148995727238103, −10.86152676639557799050269021533, −9.6756123107521542940320559055, −8.43719589198585970203671365144, −7.12767703219243478879703208242, −5.36672926330814547775328021135, −4.23827720236324856876533784183, −3.49939000225067843715739530974, −1.87963290100655522310819934115, 2.17124422313873914995038832743, 3.35543910553183348187052367311, 4.39305074199880611140651409399, 6.29343842189649275147606404570, 7.20751273212073776902243070261, 7.977929758778525413850569386860, 9.52857034448097743065054292241, 11.50754905191720469082301735464, 12.03210032891707457378808987265, 13.34985689537070506102505800328, 14.395019518163846141303100152012, 14.901249138890690277871397460732, 16.07358009505370456665528213425, 17.41258242378151080565124405605, 18.786826109111649133908990326326, 19.75417536260560036622970536249, 20.481424548307395631809344541, 21.96601947563639503122404492998, 22.68501871490264645435635864517, 23.88395085315404409190946625783, 24.48579943652112040188447923612, 25.524355132855480999577287351993, 26.42417657585609683976964993394, 27.36486898630404962445455070351, 29.23915862782626841885974256190

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