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Properties

Label 1-133-133.52-r0-0-0
Degree $1$
Conductor $133$
Sign $0.827 + 0.561i$
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  + (−0.173 − 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (0.939 − 0.342i)6-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.173 − 0.984i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.939 + 0.342i)13-s + (−0.173 + 0.984i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + (0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (0.939 − 0.342i)6-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.173 − 0.984i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.939 + 0.342i)13-s + (−0.173 + 0.984i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + (0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ 0.827 + 0.561i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9557654446 + 0.2936765015i\)
\(L(\frac12)\) \(\approx\) \(0.9557654446 + 0.2936765015i\)
\(L(1)\) \(\approx\) \(0.9963577996 + 0.06178045998i\)
\(L(1)\) \(\approx\) \(0.9963577996 + 0.06178045998i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.173 - 0.984i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
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

−28.69566696767279487205400906990, −27.32718974607449667523057414676, −26.21389276287985845823088530776, −25.33628731446749312222245053002, −24.62121337896823207531031163208, −23.940134896049647557580867010361, −22.82146513070735342484529772943, −21.697799624634260093197268518876, −20.35195809809817951899094932022, −18.9411966748257836524649471469, −18.35554321989261217353906488075, −17.179937746100743216291553451559, −16.64390771883590912923696962049, −15.0076606288418803601203236825, −14.00086236960964129898241846294, −13.29072667909132147591909807265, −12.28486747797088075883432889294, −10.37541103136168725781289643775, −9.14636502019750640559466599779, −8.16615925475926284375551775401, −7.092686683004487154639858187645, −5.93034398257344301925621545972, −5.10683121380573978616459287989, −2.831325234310403696364763798164, −1.0186674231738255900931812885, 2.04233867030937510828243043273, 3.10292537073743046572638085804, 4.55723163298923090479806019221, 5.55636308096774975820853461454, 7.62555231337173420128514444929, 9.18502524720484159878663906138, 9.87798747080487215958154652989, 10.58027292376744972862767323924, 11.862121912784664031343354196548, 13.14244985531570004787666222604, 14.25401419695295009422131788611, 15.10822577356755637317686861047, 16.86096610803125583903132518081, 17.46279856275876530065955903923, 18.73306300463431334651197446021, 19.81544793452275516619301885200, 20.95272586320526158192793618845, 21.394391832172534631290448410736, 22.3793674072769634564334325050, 23.220435352653463355710649551366, 25.14136702945597988470584255517, 26.09224077671414689378289743072, 26.74545644790270376256131323518, 27.90153936042004896008102876749, 28.63774294666065798851912586354

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