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Properties

Label 1-199-199.9-r0-0-0
Degree $1$
Conductor $199$
Sign $0.237 + 0.971i$
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  + (0.902 + 0.429i)2-s + (0.997 + 0.0634i)3-s + (0.630 + 0.776i)4-s + (−0.786 + 0.618i)5-s + (0.873 + 0.486i)6-s + (−0.916 + 0.400i)7-s + (0.235 + 0.971i)8-s + (0.991 + 0.126i)9-s + (−0.975 + 0.220i)10-s + (0.841 − 0.540i)11-s + (0.580 + 0.814i)12-s + (−0.0792 − 0.996i)13-s + (−0.999 − 0.0317i)14-s + (−0.823 + 0.567i)15-s + (−0.204 + 0.978i)16-s + (0.0475 + 0.998i)17-s + ⋯
L(s)  = 1  + (0.902 + 0.429i)2-s + (0.997 + 0.0634i)3-s + (0.630 + 0.776i)4-s + (−0.786 + 0.618i)5-s + (0.873 + 0.486i)6-s + (−0.916 + 0.400i)7-s + (0.235 + 0.971i)8-s + (0.991 + 0.126i)9-s + (−0.975 + 0.220i)10-s + (0.841 − 0.540i)11-s + (0.580 + 0.814i)12-s + (−0.0792 − 0.996i)13-s + (−0.999 − 0.0317i)14-s + (−0.823 + 0.567i)15-s + (−0.204 + 0.978i)16-s + (0.0475 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.237 + 0.971i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.237 + 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.771585541 + 1.390127615i\)
\(L(\frac12)\) \(\approx\) \(1.771585541 + 1.390127615i\)
\(L(1)\) \(\approx\) \(1.752455288 + 0.8299787075i\)
\(L(1)\) \(\approx\) \(1.752455288 + 0.8299787075i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.902 + 0.429i)T \)
3 \( 1 + (0.997 + 0.0634i)T \)
5 \( 1 + (-0.786 + 0.618i)T \)
7 \( 1 + (-0.916 + 0.400i)T \)
11 \( 1 + (0.841 - 0.540i)T \)
13 \( 1 + (-0.0792 - 0.996i)T \)
17 \( 1 + (0.0475 + 0.998i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.356 + 0.934i)T \)
29 \( 1 + (-0.266 - 0.963i)T \)
31 \( 1 + (-0.553 - 0.832i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (-0.386 - 0.922i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.110 - 0.993i)T \)
53 \( 1 + (-0.823 - 0.567i)T \)
59 \( 1 + (0.981 - 0.189i)T \)
61 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (-0.999 + 0.0317i)T \)
73 \( 1 + (-0.987 - 0.158i)T \)
79 \( 1 + (-0.745 - 0.666i)T \)
83 \( 1 + (0.580 - 0.814i)T \)
89 \( 1 + (-0.701 + 0.712i)T \)
97 \( 1 + (-0.444 + 0.895i)T \)
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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

−26.75472110448609527685079087879, −25.46775511780867991521366475492, −24.81313100826560755304641903703, −23.75909460416024609636248338090, −23.03267225247306744593854387590, −21.94375576020879795312291751398, −20.80610691600655081239890537372, −20.0892474540974593372945963068, −19.46771279119567783305197346129, −18.713688996653758803814632146389, −16.56195311740958294587995959337, −15.979563153500889262894350079301, −14.74922848400262497363247761559, −14.12316295053067046579066295, −12.8238619724180073452999918323, −12.43605356408932907755336585225, −11.1115173605932854553154612431, −9.695671072310069918932900788687, −8.955330307484869704678999943, −7.31479185895284324674425873947, −6.58312515852830287347252498905, −4.5926870836219908753048882399, −3.98946997058875145840635365517, −2.87644281693909305664449653708, −1.4021080617512821882519498169, 2.41552535701460553025006912137, 3.46150354760066724318971797559, 4.02352760757681763418482388138, 5.872556376993270716562904165573, 6.88581632495041205924087602722, 7.91158299210833021514559350904, 8.84880458931945326520966887251, 10.35503548283265840220882395935, 11.61352721183604649495256767501, 12.76771661746329902808720131760, 13.46500825812988277027047720562, 14.82591263497567599595573333098, 15.14932348366941339603906488225, 16.03832636856913810256511967399, 17.2652988776832451993270558975, 18.99545097141181920939517203027, 19.486837042820409836388014673878, 20.456484745791705370670435811666, 21.78853230963788249280699746420, 22.24691658540314489718819919207, 23.373145802443385806385690280101, 24.29513252159009927197914937069, 25.381535983316742323737886221464, 25.827382304561182464605162763599, 26.84502790195493129299958223848

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