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Properties

Label 1-199-199.4-r0-0-0
Degree $1$
Conductor $199$
Sign $0.967 + 0.253i$
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  + (−0.999 + 0.0317i)2-s + (0.902 + 0.429i)3-s + (0.997 − 0.0634i)4-s + (0.0475 − 0.998i)5-s + (−0.916 − 0.400i)6-s + (0.967 + 0.251i)7-s + (−0.995 + 0.0950i)8-s + (0.630 + 0.776i)9-s + (−0.0158 + 0.999i)10-s + (−0.654 + 0.755i)11-s + (0.928 + 0.371i)12-s + (0.527 + 0.849i)13-s + (−0.975 − 0.220i)14-s + (0.472 − 0.881i)15-s + (0.991 − 0.126i)16-s + (−0.327 − 0.945i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0317i)2-s + (0.902 + 0.429i)3-s + (0.997 − 0.0634i)4-s + (0.0475 − 0.998i)5-s + (−0.916 − 0.400i)6-s + (0.967 + 0.251i)7-s + (−0.995 + 0.0950i)8-s + (0.630 + 0.776i)9-s + (−0.0158 + 0.999i)10-s + (−0.654 + 0.755i)11-s + (0.928 + 0.371i)12-s + (0.527 + 0.849i)13-s + (−0.975 − 0.220i)14-s + (0.472 − 0.881i)15-s + (0.991 − 0.126i)16-s + (−0.327 − 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.967 + 0.253i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.967 + 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.123701207 + 0.1449568811i\)
\(L(\frac12)\) \(\approx\) \(1.123701207 + 0.1449568811i\)
\(L(1)\) \(\approx\) \(1.016574468 + 0.08229583081i\)
\(L(1)\) \(\approx\) \(1.016574468 + 0.08229583081i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.999 + 0.0317i)T \)
3 \( 1 + (0.902 + 0.429i)T \)
5 \( 1 + (0.0475 - 0.998i)T \)
7 \( 1 + (0.967 + 0.251i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (0.527 + 0.849i)T \)
17 \( 1 + (-0.327 - 0.945i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.553 + 0.832i)T \)
29 \( 1 + (0.950 - 0.312i)T \)
31 \( 1 + (-0.823 - 0.567i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.356 - 0.934i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.701 + 0.712i)T \)
53 \( 1 + (0.472 + 0.881i)T \)
59 \( 1 + (0.235 - 0.971i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.975 + 0.220i)T \)
73 \( 1 + (-0.444 - 0.895i)T \)
79 \( 1 + (-0.386 + 0.922i)T \)
83 \( 1 + (0.928 - 0.371i)T \)
89 \( 1 + (-0.745 - 0.666i)T \)
97 \( 1 + (-0.0792 + 0.996i)T \)
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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

−26.803603529747221177292704243414, −26.08589733173784819249127797604, −25.24251105970203850347250184877, −24.30041545524477656247620857635, −23.493830458720763663473389839772, −21.793478283564374319578180905194, −20.87842496300454060790450098409, −20.07595075103285445268418415143, −19.09594366754299948396856221274, −18.09090713010966156994782461239, −17.95027295545948120841871731242, −16.26147676544800317927845399356, −15.15315270882776568497969767673, −14.46170793964307319750256556231, −13.34120544732374671423368600201, −11.91208022880500853140595596234, −10.71303631405402388879613115179, −10.16041442976502533987678651789, −8.46485563046605092275641727612, −8.089524994301669934329239403998, −7.031913688574609195216398103392, −5.87922707791130505519247735886, −3.582342687516643358722068407038, −2.60222652138141741565503697546, −1.37028231593528912446213995072, 1.516754438013977791733211162805, 2.49200204235005415924310607381, 4.304550753454264236142222357788, 5.392224233415613662956063513249, 7.33097443158196599098461609519, 8.07119419064842218081428690701, 9.10537737542456735359042960242, 9.594827335219170642275812035840, 11.01185198420471006289110475818, 11.99295657129273044571791036635, 13.39264831034193217982544752466, 14.496717364153183288946408454816, 15.78335099801641492821210402761, 16.05735275187075454317025672196, 17.53619880491631688005021117054, 18.23469875053011735053107001168, 19.46909358767195348264497205783, 20.38059318161284115741262507506, 20.856011250351883817386882434973, 21.65418742786138633409886679598, 23.667848227112437268593184870631, 24.46391149487488906091081433235, 25.20065872492593482906485586763, 26.09015661361016576707058744526, 26.95011574185592759579381655646

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