Properties

Label 1-199-199.35-r0-0-0
Degree $1$
Conductor $199$
Sign $0.848 - 0.529i$
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  + (0.805 − 0.592i)2-s + (−0.857 + 0.513i)3-s + (0.296 − 0.954i)4-s + (0.580 + 0.814i)5-s + (−0.386 + 0.922i)6-s + (0.356 − 0.934i)7-s + (−0.327 − 0.945i)8-s + (0.472 − 0.881i)9-s + (0.950 + 0.312i)10-s + (−0.142 + 0.989i)11-s + (0.235 + 0.971i)12-s + (0.110 + 0.993i)13-s + (−0.266 − 0.963i)14-s + (−0.916 − 0.400i)15-s + (−0.823 − 0.567i)16-s + (0.928 − 0.371i)17-s + ⋯
L(s)  = 1  + (0.805 − 0.592i)2-s + (−0.857 + 0.513i)3-s + (0.296 − 0.954i)4-s + (0.580 + 0.814i)5-s + (−0.386 + 0.922i)6-s + (0.356 − 0.934i)7-s + (−0.327 − 0.945i)8-s + (0.472 − 0.881i)9-s + (0.950 + 0.312i)10-s + (−0.142 + 0.989i)11-s + (0.235 + 0.971i)12-s + (0.110 + 0.993i)13-s + (−0.266 − 0.963i)14-s + (−0.916 − 0.400i)15-s + (−0.823 − 0.567i)16-s + (0.928 − 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.848 - 0.529i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.848 - 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.529950458 - 0.4386404830i\)
\(L(\frac12)\) \(\approx\) \(1.529950458 - 0.4386404830i\)
\(L(1)\) \(\approx\) \(1.385899926 - 0.2881451238i\)
\(L(1)\) \(\approx\) \(1.385899926 - 0.2881451238i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.805 - 0.592i)T \)
3 \( 1 + (-0.857 + 0.513i)T \)
5 \( 1 + (0.580 + 0.814i)T \)
7 \( 1 + (0.356 - 0.934i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (0.110 + 0.993i)T \)
17 \( 1 + (0.928 - 0.371i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.678 - 0.734i)T \)
29 \( 1 + (0.997 - 0.0634i)T \)
31 \( 1 + (0.873 - 0.486i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (0.527 + 0.849i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.987 + 0.158i)T \)
53 \( 1 + (-0.916 + 0.400i)T \)
59 \( 1 + (0.0475 - 0.998i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (-0.266 + 0.963i)T \)
73 \( 1 + (-0.975 - 0.220i)T \)
79 \( 1 + (-0.0792 + 0.996i)T \)
83 \( 1 + (0.235 - 0.971i)T \)
89 \( 1 + (-0.444 + 0.895i)T \)
97 \( 1 + (-0.0158 + 0.999i)T \)
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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

−27.14566556902218457358865928476, −25.38013809872862307467542782600, −24.98393963662587022324431291038, −24.20338503355214645520740336563, −23.35482299162215813375565762421, −22.380116143417498041199946873527, −21.40853642635467089060376553551, −20.934881626493048530975375360430, −19.21738351729154765708334980137, −18.01097635492212201811278956973, −17.27926296639371729355680918950, −16.36837789246570930781261962327, −15.57319891974087558286232201904, −14.17689040989810259486530345067, −13.241325805033384955640644532712, −12.386204683397315982634996394408, −11.750289195738186379002576789479, −10.35810350416883198781273255787, −8.57050037031724306350445119550, −7.886387749680636141250813476750, −6.257391581931535791298255284123, −5.56166375622247940598958633112, −4.99761295049153929334624539313, −3.1340198702023655515224628213, −1.50169273959635051046456234333, 1.36831340837636826500425721525, 2.9353078754543692695384400173, 4.32840969516930908010498010468, 5.03010305759403676108208581146, 6.46661039091299543304440598408, 7.074080907254230607009519577028, 9.58074769630026879542499374920, 10.21036693028137665919190623281, 11.08953384910328888515089574685, 11.88159631207003006361011998951, 13.15145847250000768683170918554, 14.20507972565201334735760784133, 14.929201253033133381903378675604, 16.11994450164015846631569303909, 17.303536871958305968843138542694, 18.157783149958283825948080194233, 19.28249242643531460324888060619, 20.675579469890086636567403178033, 21.17907469756610300833547719186, 22.13424869775942905254720142909, 23.07745261821595016788290761403, 23.41247150990148533506040103838, 24.682148997391603096759508270438, 26.09139168715008823492559012410, 26.90977151727635428475117075752

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