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Properties

Label 1-199-199.18-r0-0-0
Degree $1$
Conductor $199$
Sign $0.984 - 0.173i$
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  + (0.415 − 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.142 + 0.989i)5-s + (−0.654 + 0.755i)6-s + (−0.959 + 0.281i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (0.841 + 0.540i)10-s + (0.841 + 0.540i)11-s + (0.415 + 0.909i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)15-s + (−0.142 + 0.989i)16-s + (0.841 + 0.540i)17-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.142 + 0.989i)5-s + (−0.654 + 0.755i)6-s + (−0.959 + 0.281i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (0.841 + 0.540i)10-s + (0.841 + 0.540i)11-s + (0.415 + 0.909i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)15-s + (−0.142 + 0.989i)16-s + (0.841 + 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8302083086 - 0.07254687836i\)
\(L(\frac12)\) \(\approx\) \(0.8302083086 - 0.07254687836i\)
\(L(1)\) \(\approx\) \(0.8165230494 - 0.2269534830i\)
\(L(1)\) \(\approx\) \(0.8165230494 - 0.2269534830i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.415 - 0.909i)T \)
3 \( 1 + (-0.959 - 0.281i)T \)
5 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.654 + 0.755i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.959 + 0.281i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.654 + 0.755i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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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

−26.931856479540722212668363692795, −25.97271249949137367041269768946, −24.76401696049177640765469900102, −24.09577245504964587952542439879, −23.185535408861554672400652087884, −22.47748248319330037501669053565, −21.56382293321418596549625473179, −20.5841578975362029413753757068, −19.13124253889215742705378974760, −17.99075977911981728729743932409, −16.74156301817591518299574510560, −16.42628629788230394723352904598, −15.84250975373679478967665148218, −14.27713643465300803214617416248, −13.28224302495103732549407047176, −12.27077736210799424960607698773, −11.57696984544598221815468495241, −9.720354101177465462687747858334, −9.07495337525108565508451455166, −7.569159601999471676633930850648, −6.38509042905335445574150486321, −5.68108584010187267320099100982, −4.42021710379513583874132846185, −3.64794883522352370781800816429, −0.74594835428747155767573093639, 1.33490853348510655818055524276, 2.96313868929861594974354038139, 3.93623544930988888242161097204, 5.58638801967689418844126530630, 6.26068193738451359890006032626, 7.51943131476351625290200515553, 9.50848479092748286552487355489, 10.268968109108185789385111614145, 11.20627658479920337394408590949, 12.15404657804585498906226564288, 12.84762385825144758353536892475, 14.05341065371187200463303254824, 15.15066679202665620657135983533, 16.22289263735681028444517975958, 17.75526270781519589377269677577, 18.29389446924825906741975870005, 19.30653306496771829396672498770, 20.02698932634931107983787015745, 21.668055861696655816734819570908, 22.19353029476869370873588230904, 22.94177587140700440371818211121, 23.47949070145203982316979336302, 24.91606075983738808389898798086, 25.99464279920568594239036499383, 27.42499203540439930844363255884

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