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Properties

Label 1-133-133.59-r0-0-0
Degree $1$
Conductor $133$
Sign $0.437 - 0.899i$
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  + (−0.766 + 0.642i)2-s + (0.766 − 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.173 + 0.984i)6-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (0.766 + 0.642i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)13-s + (−0.766 − 0.642i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.766 − 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.173 + 0.984i)6-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (0.766 + 0.642i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)13-s + (−0.766 − 0.642i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ 0.437 - 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7304388418 - 0.4570063137i\)
\(L(\frac12)\) \(\approx\) \(0.7304388418 - 0.4570063137i\)
\(L(1)\) \(\approx\) \(0.8484550623 - 0.2061516794i\)
\(L(1)\) \(\approx\) \(0.8484550623 - 0.2061516794i\)

Euler product

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

−28.68695002517213141185869127918, −27.64902763114151625836676646842, −26.47736002717050469993060288255, −26.41416775488168987292531323621, −25.34606529159045553721380313869, −23.89378793599986626484999323524, −22.24636048281969943174030157969, −21.601267021657753677935920148549, −20.76846039875575863703100706622, −19.41243011299455079937398849343, −19.05301602138547783605158245178, −17.891049316654075122155086516119, −16.46704936790884811078547089700, −15.65106640106129314481369256547, −14.34153668139799832430298215837, −13.381521611709030378458425784670, −11.73520788739013841752780099652, −10.69958092226543601939957936395, −10.02456981422393989865215188086, −8.68825327510458225809583154832, −7.90928601116273103686078479562, −6.48162592793954999743313600485, −4.21470781046753789125870675003, −3.20220396557323551966891742074, −2.123861997389562748820656866246, 0.96030141824756880495517419065, 2.46141894578603389004761104653, 4.58483813553486446330083380727, 5.96656363997272870387041751419, 7.459087624461769129777871031170, 8.07392483427464086439557925224, 9.15661920826201547126157316815, 10.090778068062167565233585700604, 11.8882151580764025235839138486, 13.02842773282119756901545685311, 14.05922260717502990746304508585, 15.36871336724365201160945361420, 15.999454438211881764562874550712, 17.551133611855775449122025982, 18.06061479880867284208135542229, 19.38523935157218485689341455714, 20.17598800875553282106753680028, 20.814959325639346293840421999495, 22.976099254828700084165140124480, 23.75760931298049017000878439261, 24.84655761552649559185750041999, 25.22375991737112725485560893547, 26.3166233204475923956896143731, 27.369277376266445310614330113673, 28.31901571272632617144601996926

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