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Properties

Label 1-199-199.184-r0-0-0
Degree $1$
Conductor $199$
Sign $0.865 + 0.501i$
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  + (−0.857 + 0.513i)2-s + (0.296 + 0.954i)3-s + (0.472 − 0.881i)4-s + (0.723 − 0.690i)5-s + (−0.745 − 0.666i)6-s + (−0.386 − 0.922i)7-s + (0.0475 + 0.998i)8-s + (−0.823 + 0.567i)9-s + (−0.266 + 0.963i)10-s + (0.415 + 0.909i)11-s + (0.981 + 0.189i)12-s + (−0.0158 − 0.999i)13-s + (0.805 + 0.592i)14-s + (0.873 + 0.486i)15-s + (−0.553 − 0.832i)16-s + (0.580 − 0.814i)17-s + ⋯
L(s)  = 1  + (−0.857 + 0.513i)2-s + (0.296 + 0.954i)3-s + (0.472 − 0.881i)4-s + (0.723 − 0.690i)5-s + (−0.745 − 0.666i)6-s + (−0.386 − 0.922i)7-s + (0.0475 + 0.998i)8-s + (−0.823 + 0.567i)9-s + (−0.266 + 0.963i)10-s + (0.415 + 0.909i)11-s + (0.981 + 0.189i)12-s + (−0.0158 − 0.999i)13-s + (0.805 + 0.592i)14-s + (0.873 + 0.486i)15-s + (−0.553 − 0.832i)16-s + (0.580 − 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.865 + 0.501i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.865 + 0.501i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9229086737 + 0.2482651541i\)
\(L(\frac12)\) \(\approx\) \(0.9229086737 + 0.2482651541i\)
\(L(1)\) \(\approx\) \(0.8583474250 + 0.2253384379i\)
\(L(1)\) \(\approx\) \(0.8583474250 + 0.2253384379i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.857 + 0.513i)T \)
3 \( 1 + (0.296 + 0.954i)T \)
5 \( 1 + (0.723 - 0.690i)T \)
7 \( 1 + (-0.386 - 0.922i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (-0.0158 - 0.999i)T \)
17 \( 1 + (0.580 - 0.814i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.527 - 0.849i)T \)
29 \( 1 + (0.630 + 0.776i)T \)
31 \( 1 + (0.678 + 0.734i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + (-0.0792 - 0.996i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.605 + 0.795i)T \)
53 \( 1 + (0.873 - 0.486i)T \)
59 \( 1 + (-0.786 + 0.618i)T \)
61 \( 1 + (-0.959 + 0.281i)T \)
67 \( 1 + (0.235 + 0.971i)T \)
71 \( 1 + (0.805 - 0.592i)T \)
73 \( 1 + (-0.999 - 0.0317i)T \)
79 \( 1 + (-0.444 + 0.895i)T \)
83 \( 1 + (0.981 - 0.189i)T \)
89 \( 1 + (-0.987 + 0.158i)T \)
97 \( 1 + (-0.975 + 0.220i)T \)
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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

−26.44939766732331413442936854277, −26.12146955156355301939169315867, −25.01145934462739663094054463859, −24.52979792386925827082661064118, −22.981173309254919738226974586261, −21.6489192162606231818418100959, −21.35134432171481309812065552057, −19.66401542838060605609617921367, −19.08815069266708901396414163136, −18.45027014767272015254068542348, −17.557578898630212592623962481620, −16.6122398084324009849847673992, −15.19170331814819204757472630904, −13.93804969395438533643502076122, −13.08579032517797163799132260615, −11.86709455600971703680196358185, −11.233881929940342517627633029323, −9.6578795872987224548268579123, −9.017876119893080438915236299055, −7.87360365522974826089785900573, −6.64721513308420354312781133121, −5.965061242267952911073338106432, −3.35704363735737657712786643943, −2.51609398314242028263670667624, −1.391057842746801144647805369387, 1.14133078729219785951815772973, 2.9116174353735668224720000253, 4.63459676366383185603309243892, 5.518245251790145736086795444804, 6.92515992740906156606237697912, 8.12909786872897206281557162597, 9.20547433060409708905138840546, 10.03463450247203263016904798020, 10.4871807094828806404780411742, 12.18744389419271503930266560216, 13.73649276919268358994642221763, 14.492775806713979408777949956673, 15.6794583983234940622715179282, 16.51632652546219486213068304343, 17.153936214936364993856384222796, 18.06877551275656428662445641323, 19.59950369525391545768852938665, 20.457055907124741189760782443513, 20.72527320591177368438202639571, 22.45124638146591770904009932839, 23.14468581321617548616566634422, 24.63737392839939980250707232482, 25.36127793578411097974751868299, 25.94871183908936916870083852803, 27.18213328300003888380900363862

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