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Properties

Label 1-199-199.40-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.471 + 0.882i$
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  + (0.723 + 0.690i)2-s + (−0.327 + 0.945i)3-s + (0.0475 + 0.998i)4-s + (0.415 − 0.909i)5-s + (−0.888 + 0.458i)6-s + (0.981 + 0.189i)7-s + (−0.654 + 0.755i)8-s + (−0.786 − 0.618i)9-s + (0.928 − 0.371i)10-s + (−0.142 + 0.989i)11-s + (−0.959 − 0.281i)12-s + (0.723 + 0.690i)13-s + (0.580 + 0.814i)14-s + (0.723 + 0.690i)15-s + (−0.995 + 0.0950i)16-s + (−0.142 + 0.989i)17-s + ⋯
L(s)  = 1  + (0.723 + 0.690i)2-s + (−0.327 + 0.945i)3-s + (0.0475 + 0.998i)4-s + (0.415 − 0.909i)5-s + (−0.888 + 0.458i)6-s + (0.981 + 0.189i)7-s + (−0.654 + 0.755i)8-s + (−0.786 − 0.618i)9-s + (0.928 − 0.371i)10-s + (−0.142 + 0.989i)11-s + (−0.959 − 0.281i)12-s + (0.723 + 0.690i)13-s + (0.580 + 0.814i)14-s + (0.723 + 0.690i)15-s + (−0.995 + 0.0950i)16-s + (−0.142 + 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.471 + 0.882i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.471 + 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8591847132 + 1.433081890i\)
\(L(\frac12)\) \(\approx\) \(0.8591847132 + 1.433081890i\)
\(L(1)\) \(\approx\) \(1.139481735 + 0.9493072197i\)
\(L(1)\) \(\approx\) \(1.139481735 + 0.9493072197i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.723 + 0.690i)T \)
3 \( 1 + (-0.327 + 0.945i)T \)
5 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (0.981 + 0.189i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (0.723 + 0.690i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.0475 - 0.998i)T \)
29 \( 1 + (0.235 + 0.971i)T \)
31 \( 1 + (-0.327 - 0.945i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.786 - 0.618i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.981 + 0.189i)T \)
53 \( 1 + (0.723 - 0.690i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (0.580 - 0.814i)T \)
73 \( 1 + (0.0475 - 0.998i)T \)
79 \( 1 + (-0.995 + 0.0950i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (-0.327 - 0.945i)T \)
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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

−26.835539647317915339947707607221, −25.205250443929543302586614237242, −24.66623760743414799667517348789, −23.37485529178124128173853748413, −23.07806345975329028709892493816, −21.8506159925564080092059095930, −21.0878225334504648177536919996, −19.96355939714315754866422201930, −18.80168862431997035663643884737, −18.30683332926427895553243288772, −17.3733988629148944035732389514, −15.741415077995268296368367967881, −14.40030941758823462909120128549, −13.86024761108275680592602772531, −13.04978166098568154889201747816, −11.631954778763897818182520914553, −11.14652502349580174016508935282, −10.22610737602438293746136396894, −8.45974477288228197983168550347, −7.196719159572716670332296291951, −6.00917776943055497378926052159, −5.31888556206054917319006173, −3.528478885732162938529877349849, −2.38056820145551517233760972524, −1.20369724351614532799640569809, 2.08548120876618359096579980385, 4.04197521491818145473489149895, 4.70485870462494543758460136379, 5.53577959631286249114518258126, 6.725994961820727801188958235792, 8.43775499252030054447562795951, 8.93724243736441992052257075510, 10.4968258203332730689098442956, 11.67980859649047310048487716348, 12.58518556661827227134585231209, 13.74731832731238604840905413704, 14.85934363149103628705083480430, 15.47579632015575221606977953429, 16.65074523871230924866206097026, 17.21992755726167610070210943045, 18.09460814652682567868365276463, 20.260840625531855384319483069465, 20.85801403912391061815462869279, 21.564081961984930982218592453806, 22.403712233487993131111767846837, 23.79332461825217959746298467302, 23.9792047475238366067656190509, 25.49097288359352619270515585356, 25.96973211483869837125732546149, 27.238716625516968755084889182762

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