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Properties

Label 1-199-199.16-r0-0-0
Degree $1$
Conductor $199$
Sign $0.765 + 0.643i$
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  + (0.997 − 0.0634i)2-s + (0.630 + 0.776i)3-s + (0.991 − 0.126i)4-s + (−0.995 − 0.0950i)5-s + (0.678 + 0.734i)6-s + (0.873 + 0.486i)7-s + (0.981 − 0.189i)8-s + (−0.204 + 0.978i)9-s + (−0.999 − 0.0317i)10-s + (−0.142 − 0.989i)11-s + (0.723 + 0.690i)12-s + (−0.444 + 0.895i)13-s + (0.902 + 0.429i)14-s + (−0.553 − 0.832i)15-s + (0.967 − 0.251i)16-s + (−0.786 + 0.618i)17-s + ⋯
L(s)  = 1  + (0.997 − 0.0634i)2-s + (0.630 + 0.776i)3-s + (0.991 − 0.126i)4-s + (−0.995 − 0.0950i)5-s + (0.678 + 0.734i)6-s + (0.873 + 0.486i)7-s + (0.981 − 0.189i)8-s + (−0.204 + 0.978i)9-s + (−0.999 − 0.0317i)10-s + (−0.142 − 0.989i)11-s + (0.723 + 0.690i)12-s + (−0.444 + 0.895i)13-s + (0.902 + 0.429i)14-s + (−0.553 − 0.832i)15-s + (0.967 − 0.251i)16-s + (−0.786 + 0.618i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.765 + 0.643i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.765 + 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.194962610 + 0.7994070419i\)
\(L(\frac12)\) \(\approx\) \(2.194962610 + 0.7994070419i\)
\(L(1)\) \(\approx\) \(1.947266415 + 0.4446449975i\)
\(L(1)\) \(\approx\) \(1.947266415 + 0.4446449975i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.997 - 0.0634i)T \)
3 \( 1 + (0.630 + 0.776i)T \)
5 \( 1 + (-0.995 - 0.0950i)T \)
7 \( 1 + (0.873 + 0.486i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (-0.444 + 0.895i)T \)
17 \( 1 + (-0.786 + 0.618i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.386 - 0.922i)T \)
29 \( 1 + (0.805 - 0.592i)T \)
31 \( 1 + (0.356 + 0.934i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.745 - 0.666i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.0158 - 0.999i)T \)
53 \( 1 + (-0.553 + 0.832i)T \)
59 \( 1 + (-0.888 - 0.458i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (0.902 - 0.429i)T \)
73 \( 1 + (-0.605 + 0.795i)T \)
79 \( 1 + (-0.701 - 0.712i)T \)
83 \( 1 + (0.723 - 0.690i)T \)
89 \( 1 + (0.110 + 0.993i)T \)
97 \( 1 + (-0.987 - 0.158i)T \)
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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

−26.70412963849048575068186178552, −25.53940143343001487589025944042, −24.71047879566396834719691490744, −23.9100315149276181518550058965, −23.211021215964615014211795414164, −22.40685302297393795851335926334, −20.799804882462714554969740542186, −20.23922807919356346832290122984, −19.60794523160913665190387501080, −18.233942154628538092869707333247, −17.23953521443205279141723057542, −15.68572440555664080607076783438, −14.99184047660224620426333572931, −14.19696830282623219371208141593, −13.166733983346859805050418753867, −12.17551303523505100025121168857, −11.52074022601008147705276674675, −10.16051962367257092506042058047, −8.19207798772213695856973196033, −7.59023867359426811020443569545, −6.789651615215075295699156181673, −5.10441045100036293055395643531, −4.04266074988068855336949337654, −2.9006776647611587417098144649, −1.57829098764212746241486316868, 2.13662439257477460955686202089, 3.30245586095563610453831361093, 4.42664117727831408184822952955, 5.04106911839957297224576908007, 6.69410153969482821415267493680, 8.08884465622643231290980514785, 8.798118165891996184750508407652, 10.59141203831485396364909362486, 11.32808789598813948493827419231, 12.20713753624778770593987061300, 13.67357347884911800238626850167, 14.4103149091321518007885799092, 15.38673224131422469030336289911, 15.89111062112725587673635013991, 16.99670292740855863921254201741, 18.88979982378702095235399870376, 19.67457095630781344511151260206, 20.50160291075999239083487969327, 21.64140933987890931388716751309, 21.86213742174653563862698745072, 23.2839202820145247953152755875, 24.29785880208767481157885730123, 24.686135170559444708236444641969, 26.29369191700897390696610830653, 26.80811065904418920724165389223

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