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Properties

Label 1-199-199.188-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: $\chi_{199} (188, \cdot )$
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

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

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