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Properties

Label 1-199-199.132-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.698 + 0.715i$
Analytic cond. $0.924152$
Root an. cond. $0.924152$
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  + (0.235 + 0.971i)2-s + (0.981 + 0.189i)3-s + (−0.888 + 0.458i)4-s + (0.415 + 0.909i)5-s + (0.0475 + 0.998i)6-s + (−0.327 + 0.945i)7-s + (−0.654 − 0.755i)8-s + (0.928 + 0.371i)9-s + (−0.786 + 0.618i)10-s + (−0.142 − 0.989i)11-s + (−0.959 + 0.281i)12-s + (0.235 + 0.971i)13-s + (−0.995 − 0.0950i)14-s + (0.235 + 0.971i)15-s + (0.580 − 0.814i)16-s + (−0.142 − 0.989i)17-s + ⋯
L(s)  = 1  + (0.235 + 0.971i)2-s + (0.981 + 0.189i)3-s + (−0.888 + 0.458i)4-s + (0.415 + 0.909i)5-s + (0.0475 + 0.998i)6-s + (−0.327 + 0.945i)7-s + (−0.654 − 0.755i)8-s + (0.928 + 0.371i)9-s + (−0.786 + 0.618i)10-s + (−0.142 − 0.989i)11-s + (−0.959 + 0.281i)12-s + (0.235 + 0.971i)13-s + (−0.995 − 0.0950i)14-s + (0.235 + 0.971i)15-s + (0.580 − 0.814i)16-s + (−0.142 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.698 + 0.715i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.698 + 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6379555674 + 1.515575579i\)
\(L(\frac12)\) \(\approx\) \(0.6379555674 + 1.515575579i\)
\(L(1)\) \(\approx\) \(1.026919625 + 1.034949105i\)
\(L(1)\) \(\approx\) \(1.026919625 + 1.034949105i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.235 + 0.971i)T \)
3 \( 1 + (0.981 + 0.189i)T \)
5 \( 1 + (0.415 + 0.909i)T \)
7 \( 1 + (-0.327 + 0.945i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (0.235 + 0.971i)T \)
17 \( 1 + (-0.142 - 0.989i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.888 - 0.458i)T \)
29 \( 1 + (0.723 + 0.690i)T \)
31 \( 1 + (0.981 - 0.189i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.928 + 0.371i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.327 + 0.945i)T \)
53 \( 1 + (0.235 - 0.971i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (-0.995 + 0.0950i)T \)
73 \( 1 + (-0.888 - 0.458i)T \)
79 \( 1 + (0.580 - 0.814i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (0.723 + 0.690i)T \)
97 \( 1 + (0.981 - 0.189i)T \)
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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

−26.580762796557586337807630578857, −25.69350307343630593611654296443, −24.69568092059387144265347728935, −23.59406026198137424570094787636, −22.82853979075182345948323270278, −21.33510581222979558865789506134, −20.782549504823592265874408220971, −19.89455508034234708961738680748, −19.511922414846646251776619757885, −18.002988420115409001758241099468, −17.28927130694785268874201601031, −15.72196554377540386513145662714, −14.563083041172202618149324809096, −13.57100929776398836325385446204, −12.912425321041344647944626002318, −12.22739298029004661728477898759, −10.23315788411816932654990450427, −10.00814563004460205130042498971, −8.65971244119472048126454498292, −7.80149855377048553052305207770, −6.034582052027931478998635844807, −4.47841279871581797218199871914, −3.73795044119077750868928608287, −2.252436595717071122058509554013, −1.19175270583721709129525393479, 2.46661879074382370999743556342, 3.3484068225207910672909998008, 4.77086831342676772688859711617, 6.19994551155523320342261423333, 6.93683908195774146294944680194, 8.33764959724042985129934663827, 9.0554756426664106347981239268, 10.03436085805737923784206767207, 11.63269849643781361473537271720, 13.14266898714606548643854806280, 13.94077404801597840212492271867, 14.56872418758230524252179571423, 15.675580270689389666172414221315, 16.19619282395871833201360871191, 17.79402659643960929561581260942, 18.720870241123093644494571475735, 19.20023969111850181670263162979, 21.00872794747201680256245531282, 21.79923656169889132724041573037, 22.310571580905484722795324290766, 23.76395435227410814614483609945, 24.65993314181093306474666980947, 25.46624339346154699375563852783, 26.19888351429737087655000103787, 26.74236400305010080053599839840

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