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Properties

Label 1-133-133.104-r1-0-0
Degree $1$
Conductor $133$
Sign $0.755 + 0.654i$
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  + (−0.939 + 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.173 − 0.984i)6-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.939 + 0.342i)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 + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + 18-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.173 − 0.984i)6-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.939 + 0.342i)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 + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.755 + 0.654i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ 0.755 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7119567571 + 0.2655214038i\)
\(L(\frac12)\) \(\approx\) \(0.7119567571 + 0.2655214038i\)
\(L(1)\) \(\approx\) \(0.5710534627 + 0.1729115889i\)
\(L(1)\) \(\approx\) \(0.5710534627 + 0.1729115889i\)

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

−28.32404059819456049742668624207, −27.27570409654193065912994257922, −26.30995168670711440013540688547, −25.53283731043194320438236650678, −24.21632063400520167210305302943, −23.563327459047727495950386040413, −22.21711900055773309380004432899, −21.03095199441151311010085472825, −19.71000456948282030649281901029, −18.83022891370148925365189791351, −18.62169144631587755270560308855, −17.17398868524134056053223741269, −16.34209834753672330540378263482, −14.984020373876887516111756488212, −13.60244398058162587172145854693, −12.27263893366914303930570361934, −11.45964114502470013717887438577, −10.64910353355027718503358120729, −9.01110493369026957286146467735, −7.84136103028838804000983636056, −7.18571970678604242991675521383, −5.94398089795328256710640312888, −3.5572907060403551065679038534, −2.33972047592829150144265270037, −0.73879043585611188522684436741, 0.69960005453447382286767482109, 2.93672370937042111385703149862, 4.65943242899960379306645444318, 5.63650598691995842145233526315, 7.41012134689936283039529396198, 8.35035404476735873332751716780, 9.49485411571153854601577805670, 10.380252585704281500142918120197, 11.471902207062868876739967923778, 12.611798679508501537533841457933, 14.71338812198885215171808641863, 15.37542511430686526979162013533, 16.280533062975938636680157574414, 17.06535117598651650200947134862, 18.15736464581681679524491224410, 19.50836603724694071195392198143, 20.42258535149429375405459034054, 20.980726440684781602032270205513, 22.78370332407216309234525550445, 23.418560719253327113569853722458, 24.77591138271140601224650054878, 25.655177583190821269911638629551, 26.78083729247813589172739782887, 27.45634945752508634779505240403, 28.19773602105782091193023549827

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