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Properties

Label 1-133-133.2-r1-0-0
Degree $1$
Conductor $133$
Sign $0.791 - 0.611i$
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  + (−0.173 − 0.984i)2-s + (0.939 + 0.342i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)6-s + (0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.173 + 0.984i)10-s + 11-s − 12-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)15-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)17-s + (0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.939 + 0.342i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)6-s + (0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.173 + 0.984i)10-s + 11-s − 12-s + (−0.173 + 0.984i)13-s + (−0.766 − 0.642i)15-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)17-s + (0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.791 - 0.611i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ 0.791 - 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.784505902 - 0.6091686319i\)
\(L(\frac12)\) \(\approx\) \(1.784505902 - 0.6091686319i\)
\(L(1)\) \(\approx\) \(1.155739123 - 0.3687603225i\)
\(L(1)\) \(\approx\) \(1.155739123 - 0.3687603225i\)

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

−27.904871798245955306455858753005, −27.27109017438288372005843535474, −26.39377945331912218229397934414, −25.40528963000523050108114441159, −24.70237490923674631672190134089, −23.64947605004505092779403710035, −22.84762743692976539742136393092, −21.60478973385617633139820259594, −19.91096708241305368013068781232, −19.383516021445244873704586471915, −18.37765486625220709998387635578, −17.26464382788525082814686477294, −15.93121834856514741147847418238, −15.00067236132253835831959321965, −14.44321580638144619670347539302, −13.187782289603037619157259508577, −12.05143021186166499392875502902, −10.26910553192358269832843084224, −9.05921997194028984483695751176, −7.9945683677644158307546241060, −7.349793323616538681582072495, −6.100852685313961065034591341258, −4.28547193231747327749697258419, −3.253405640436247080013060638059, −0.98850182656088194457211358982, 1.12926363185869869791531241450, 2.73982463878647566870084849891, 3.93980123579579611091664937577, 4.68563694939317722221064406629, 7.202658466727568045570816160, 8.47009962295191611213699197363, 9.147767902075758987468790320284, 10.303301933370741491973737513652, 11.66744869466915142542651760877, 12.40513057000268263431623394069, 13.85380400030547502025915064071, 14.574750705459317404326131011213, 16.01970287168751897199249695983, 17.01374803733734467224206235531, 18.750282112208850844151602710879, 19.28403837683266274015892481975, 20.18693439221849259139015257881, 20.94611711283513556288313234647, 21.9929304531736628081757688899, 23.05727088099051608841222591605, 24.30286574995377285455136501398, 25.48921836388460031569036489632, 26.74446716646940732434954077531, 27.21168323221362693734266447583, 28.1025743675896670163364187051

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