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Properties

Label 1-199-199.162-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.957 - 0.287i$
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  + (0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.5 − 0.866i)5-s + (−0.939 + 0.342i)6-s + (0.766 − 0.642i)7-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.939 − 0.342i)10-s + 11-s + (−0.5 + 0.866i)12-s + (−0.939 − 0.342i)13-s + (0.173 − 0.984i)14-s + (0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.5 − 0.866i)5-s + (−0.939 + 0.342i)6-s + (0.766 − 0.642i)7-s + (−0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.939 − 0.342i)10-s + 11-s + (−0.5 + 0.866i)12-s + (−0.939 − 0.342i)13-s + (0.173 − 0.984i)14-s + (0.173 + 0.984i)15-s + (−0.939 − 0.342i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.957 - 0.287i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (162, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.957 - 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1662871224 - 1.134030520i\)
\(L(\frac12)\) \(\approx\) \(0.1662871224 - 1.134030520i\)
\(L(1)\) \(\approx\) \(0.7482270998 - 0.8203189079i\)
\(L(1)\) \(\approx\) \(0.7482270998 - 0.8203189079i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−27.16230607119503383910578295547, −26.605338866019617748348168646976, −25.19367650935956818396072720902, −24.34275266703502102968625220898, −23.479748971995040957320969287144, −22.58337232360415393436003026352, −21.92163364356529387386828245567, −21.310913953097580055838543643411, −19.807203465876529139757879295396, −18.36079131614471393613487511399, −17.59551354624715101583473380228, −16.63997245096516492207736823830, −15.596839767017851848195994500387, −14.83229842741102835136912043118, −14.10088549581748795224560203595, −12.339344463985362325854000278524, −11.76690663954923320369228203446, −11.01459528065933571287760391235, −9.43985686608496792062354174479, −7.96127781831961931486573927794, −6.83407771809683469113072495128, −6.07778337091980193119113779679, −4.767920140422151283662746074185, −4.0197247049786579164774161963, −2.40233704154323844755359381697, 0.79631908042554374724448013867, 1.97477727668274414266426877610, 4.23607177205486657987511381565, 4.548224617084707885651328985349, 5.87674522004059693924717412300, 7.01068566114126412853142725946, 8.388896740341574115406027001209, 10.00975674733539479839844738348, 10.97399776427017426688949394265, 11.916031378759301838498315483967, 12.48739990704724723136823847715, 13.53293641866444138539154815062, 14.68313269733689152897568656447, 15.796228496880074016305211792814, 17.05565515508271853629682161153, 17.61323590818976037281609444022, 19.28882922887931328404622413587, 19.75745776805017066844934650713, 20.89620083161237691658885938939, 21.86269598467189584845369018746, 22.700027359987579177411338550685, 23.78669182669381245699740886615, 24.11873619136223811977758773334, 24.960819912342792180632996973824, 27.096012039192682599357193168976

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