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Properties

Label 1-199-199.169-r0-0-0
Degree $1$
Conductor $199$
Sign $0.185 - 0.982i$
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  + (0.527 + 0.849i)2-s + (−0.0792 − 0.996i)3-s + (−0.444 + 0.895i)4-s + (0.0475 − 0.998i)5-s + (0.805 − 0.592i)6-s + (−0.266 − 0.963i)7-s + (−0.995 + 0.0950i)8-s + (−0.987 + 0.158i)9-s + (0.873 − 0.486i)10-s + (−0.654 + 0.755i)11-s + (0.928 + 0.371i)12-s + (0.472 − 0.881i)13-s + (0.678 − 0.734i)14-s + (−0.999 + 0.0317i)15-s + (−0.605 − 0.795i)16-s + (−0.327 − 0.945i)17-s + ⋯
L(s)  = 1  + (0.527 + 0.849i)2-s + (−0.0792 − 0.996i)3-s + (−0.444 + 0.895i)4-s + (0.0475 − 0.998i)5-s + (0.805 − 0.592i)6-s + (−0.266 − 0.963i)7-s + (−0.995 + 0.0950i)8-s + (−0.987 + 0.158i)9-s + (0.873 − 0.486i)10-s + (−0.654 + 0.755i)11-s + (0.928 + 0.371i)12-s + (0.472 − 0.881i)13-s + (0.678 − 0.734i)14-s + (−0.999 + 0.0317i)15-s + (−0.605 − 0.795i)16-s + (−0.327 − 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.185 - 0.982i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.185 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8081965320 - 0.6701863591i\)
\(L(\frac12)\) \(\approx\) \(0.8081965320 - 0.6701863591i\)
\(L(1)\) \(\approx\) \(1.042864079 - 0.2110070353i\)
\(L(1)\) \(\approx\) \(1.042864079 - 0.2110070353i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.527 + 0.849i)T \)
3 \( 1 + (-0.0792 - 0.996i)T \)
5 \( 1 + (0.0475 - 0.998i)T \)
7 \( 1 + (-0.266 - 0.963i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (0.472 - 0.881i)T \)
17 \( 1 + (-0.327 - 0.945i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.997 + 0.0634i)T \)
29 \( 1 + (-0.745 - 0.666i)T \)
31 \( 1 + (0.902 - 0.429i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (0.630 + 0.776i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.967 + 0.251i)T \)
53 \( 1 + (-0.999 - 0.0317i)T \)
59 \( 1 + (0.235 - 0.971i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (0.678 + 0.734i)T \)
73 \( 1 + (-0.553 + 0.832i)T \)
79 \( 1 + (0.991 - 0.126i)T \)
83 \( 1 + (0.928 - 0.371i)T \)
89 \( 1 + (-0.204 + 0.978i)T \)
97 \( 1 + (-0.823 - 0.567i)T \)
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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

−27.19448542835657897571878027632, −26.40353096933470329502157411473, −25.482757472666366979329958558631, −23.91028008257218975692116429222, −23.0034470400875928386910367449, −22.14388219889786236550666890733, −21.41283083685526443476892267680, −21.01126791120898960276254950551, −19.33302639186015350589805223485, −18.92276576209173190412379271053, −17.77008622912423332451128695632, −16.234325312058546467093988549886, −15.19976123399630539723189341938, −14.63413530252229546409377736675, −13.51049773341350017560351944255, −12.25503418429444915232064181508, −10.93836207657302513406095939815, −10.80063190916478696088261094714, −9.421367116211197315398876926519, −8.58744187322612816115553274277, −6.32297961479204887743199878116, −5.64650052033805492020336806522, −4.22408142334529302546780230727, −3.205795976096417729077306060987, −2.28054692384355967280031144104, 0.67497667005289196933044642127, 2.661256732850100260249199734053, 4.3038308262531723509467215777, 5.31184845394298808253580894040, 6.478924367450553961174506107786, 7.51515385269428746023220370025, 8.21198037276289966968869745772, 9.47338712103459626237867369122, 11.19847414049342309074053208122, 12.57001004639342514820130153392, 13.12016544373012573198227472369, 13.68280751174819660187838252240, 15.10536857652109146886080101488, 16.13406259336550766475748913472, 17.24226279051084414709504615631, 17.58877212224448446657389580613, 18.9262953453736033535414124037, 20.34740612805506271307548570084, 20.75477535214525429587319337205, 22.48930628212733639213934371575, 23.294057764106576071825321470498, 23.741786180220065234889141449151, 24.87051355001629116006315135828, 25.37313138297198717397220206868, 26.36766649636028280370990284786

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