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Properties

Label 1-199-199.157-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.768 + 0.639i$
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  + (−0.327 + 0.945i)2-s + (0.0475 − 0.998i)3-s + (−0.786 − 0.618i)4-s + (−0.959 − 0.281i)5-s + (0.928 + 0.371i)6-s + (−0.888 − 0.458i)7-s + (0.841 − 0.540i)8-s + (−0.995 − 0.0950i)9-s + (0.580 − 0.814i)10-s + (0.415 + 0.909i)11-s + (−0.654 + 0.755i)12-s + (−0.327 + 0.945i)13-s + (0.723 − 0.690i)14-s + (−0.327 + 0.945i)15-s + (0.235 + 0.971i)16-s + (0.415 + 0.909i)17-s + ⋯
L(s)  = 1  + (−0.327 + 0.945i)2-s + (0.0475 − 0.998i)3-s + (−0.786 − 0.618i)4-s + (−0.959 − 0.281i)5-s + (0.928 + 0.371i)6-s + (−0.888 − 0.458i)7-s + (0.841 − 0.540i)8-s + (−0.995 − 0.0950i)9-s + (0.580 − 0.814i)10-s + (0.415 + 0.909i)11-s + (−0.654 + 0.755i)12-s + (−0.327 + 0.945i)13-s + (0.723 − 0.690i)14-s + (−0.327 + 0.945i)15-s + (0.235 + 0.971i)16-s + (0.415 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.768 + 0.639i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.768 + 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08166151552 + 0.2259179562i\)
\(L(\frac12)\) \(\approx\) \(0.08166151552 + 0.2259179562i\)
\(L(1)\) \(\approx\) \(0.5052217279 + 0.1024562434i\)
\(L(1)\) \(\approx\) \(0.5052217279 + 0.1024562434i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.327 + 0.945i)T \)
3 \( 1 + (0.0475 - 0.998i)T \)
5 \( 1 + (-0.959 - 0.281i)T \)
7 \( 1 + (-0.888 - 0.458i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (-0.327 + 0.945i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.786 + 0.618i)T \)
29 \( 1 + (0.981 + 0.189i)T \)
31 \( 1 + (0.0475 + 0.998i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.995 - 0.0950i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.888 - 0.458i)T \)
53 \( 1 + (-0.327 - 0.945i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (-0.959 + 0.281i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (0.723 + 0.690i)T \)
73 \( 1 + (-0.786 + 0.618i)T \)
79 \( 1 + (0.235 + 0.971i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (0.981 + 0.189i)T \)
97 \( 1 + (0.0475 + 0.998i)T \)
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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

−26.88788007858343877122873535453, −26.03295948936722790223492657167, −24.93416344776964309104671764576, −23.099008420635948285318394411005, −22.56815249970039241170514548110, −21.84109654600802511370689265367, −20.75566456317043153819810469719, −19.83398182584499836695669305845, −19.19465418268481394206617945066, −18.2089023327117899665809594176, −16.69802216904130345844836824609, −16.1205657826683572876938737928, −14.96198787641013077774516964777, −13.86430611372261872217595877957, −12.3940431081483267742505531535, −11.695527521230948940649768294133, −10.60864465308578637611104620286, −9.8472196622255015914637805888, −8.73732496745313081294087023924, −7.88568563644625815243696359461, −5.98464934484231869620743988577, −4.52773046613372455613361872632, −3.4191900600104629284699148502, −2.84170480658351593656288687432, −0.21144133591513229603261317784, 1.51668742263283567571303274857, 3.64321852020897912092123470170, 4.87852831567352825628176682599, 6.59844082076831503330789291615, 6.93799924543701268898470013989, 8.067315705661135795975477091081, 8.9908232373608898907763982618, 10.22919735391807354043222146642, 11.849616857312857891673253486539, 12.70648053622249787841491177936, 13.725910522316216846909145460855, 14.75794148495265376010288605530, 15.77256327500444104865598930029, 16.83222064031155196047091848818, 17.47634985575144490964044899084, 18.76697201418311120260201254489, 19.55362701537673737757226348948, 19.92757812037981870866772320385, 22.01548936244979773519752131593, 23.173803439624840011335884140872, 23.5712247971263717578260515586, 24.32475510098973136966875699182, 25.57182881312826748333325660358, 25.98014496989164072615532012589, 27.14458864519001039945093232890

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