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Properties

Label 1-133-133.75-r0-0-0
Degree $1$
Conductor $133$
Sign $-0.991 + 0.126i$
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 − 6-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)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 − 6-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + 13-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ -0.991 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06994644265 + 1.102209881i\)
\(L(\frac12)\) \(\approx\) \(0.06994644265 + 1.102209881i\)
\(L(1)\) \(\approx\) \(0.6250419284 + 0.9134361904i\)
\(L(1)\) \(\approx\) \(0.6250419284 + 0.9134361904i\)

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 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + T \)
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 \( 1 - T \)
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 \( 1 + T \)
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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

−28.18984281405684453725963279102, −27.92753873923507663517005016263, −26.06184800662485349966278885309, −24.778357893265967858699100681023, −23.884243151991200851936857590408, −23.337280062011862029203795045908, −21.984119323895222165642059792002, −21.175070705092663507791908924432, −20.13315190602188970264254587553, −19.03911724225879348278347537517, −18.22613204391823343827176005476, −17.12848419165483975093483294351, −15.94004622108482705748428810256, −14.18149752196155915969206288914, −13.25932672253795219973511561683, −12.73552684608594221120333493644, −11.51829338210531233835116489343, −10.631556260829191561260594398393, −9.16130804140744931131766168686, −7.98814946845869622842260212715, −5.91365156205719358056582338160, −5.59499860070323794052138212909, −3.8746696063541265566814610521, −2.13541106232670446026795700486, −0.9700584322499985241850265163, 2.881182389982043602372964926282, 4.14676673837244830078946855251, 5.39480074464388858889896647614, 6.30595930958924657102658583498, 7.47164962902613013021830003009, 9.0707673919978481633055511760, 10.14872811616261898231785641696, 11.309501550307379406342106864173, 12.63384961399088348763723522317, 13.97135232325788212986940655237, 14.82496180150533177688873474180, 15.74445974185388286908659480955, 16.62618457470737216703432613009, 17.85344750275138288264263843353, 18.369811396383940945613567227249, 20.605863792870678173905042406780, 21.2579971142769457936651801042, 22.50265264189672757776551192936, 22.80345621744470141600285311899, 23.89256071453913492711367255586, 25.47109727731910088784456001908, 25.93027916492844654179215167759, 26.89356318955760497767273534342, 27.88895392476672699659188282899, 29.082718838835831223760945408886

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