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Properties

Label 1-199-199.65-r0-0-0
Degree $1$
Conductor $199$
Sign $0.939 - 0.342i$
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  + (0.967 − 0.251i)2-s + (−0.916 − 0.400i)3-s + (0.873 − 0.486i)4-s + (0.928 + 0.371i)5-s + (−0.987 − 0.158i)6-s + (−0.444 + 0.895i)7-s + (0.723 − 0.690i)8-s + (0.678 + 0.734i)9-s + (0.991 + 0.126i)10-s + (0.841 − 0.540i)11-s + (−0.995 + 0.0950i)12-s + (−0.266 + 0.963i)13-s + (−0.204 + 0.978i)14-s + (−0.701 − 0.712i)15-s + (0.527 − 0.849i)16-s + (−0.888 − 0.458i)17-s + ⋯
L(s)  = 1  + (0.967 − 0.251i)2-s + (−0.916 − 0.400i)3-s + (0.873 − 0.486i)4-s + (0.928 + 0.371i)5-s + (−0.987 − 0.158i)6-s + (−0.444 + 0.895i)7-s + (0.723 − 0.690i)8-s + (0.678 + 0.734i)9-s + (0.991 + 0.126i)10-s + (0.841 − 0.540i)11-s + (−0.995 + 0.0950i)12-s + (−0.266 + 0.963i)13-s + (−0.204 + 0.978i)14-s + (−0.701 − 0.712i)15-s + (0.527 − 0.849i)16-s + (−0.888 − 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.795837101 - 0.3171665210i\)
\(L(\frac12)\) \(\approx\) \(1.795837101 - 0.3171665210i\)
\(L(1)\) \(\approx\) \(1.577401122 - 0.2397136703i\)
\(L(1)\) \(\approx\) \(1.577401122 - 0.2397136703i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.967 - 0.251i)T \)
3 \( 1 + (-0.916 - 0.400i)T \)
5 \( 1 + (0.928 + 0.371i)T \)
7 \( 1 + (-0.444 + 0.895i)T \)
11 \( 1 + (0.841 - 0.540i)T \)
13 \( 1 + (-0.266 + 0.963i)T \)
17 \( 1 + (-0.888 - 0.458i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.0158 - 0.999i)T \)
29 \( 1 + (-0.823 - 0.567i)T \)
31 \( 1 + (0.110 - 0.993i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + (-0.975 + 0.220i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.997 - 0.0634i)T \)
53 \( 1 + (-0.701 + 0.712i)T \)
59 \( 1 + (-0.327 + 0.945i)T \)
61 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (-0.204 - 0.978i)T \)
73 \( 1 + (-0.857 + 0.513i)T \)
79 \( 1 + (-0.999 - 0.0317i)T \)
83 \( 1 + (-0.995 - 0.0950i)T \)
89 \( 1 + (0.902 - 0.429i)T \)
97 \( 1 + (0.805 + 0.592i)T \)
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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

−26.873638615123513727518211814661, −25.858209122324307220368082771047, −24.88539151643646333922313793101, −24.00334861125325987625004152038, −23.04414939236072603916993889024, −22.216713811795550985529172491263, −21.72016364601808555382801703237, −20.43990414828936314304442998053, −19.88067960519912180851625366177, −17.58862944159576635561043333281, −17.394807972814325108157254881071, −16.32526968084866362149522694968, −15.43410757949700080098166953444, −14.24934544072110562226034394526, −13.15526811671491255418121413766, −12.540860854915789509600612718528, −11.261224973990983199156943601956, −10.32595283054911196705192487068, −9.29093357525159963450166033049, −7.29680541287760042766061431236, −6.46730122525148188709188640856, −5.44522255596641172138093978568, −4.551340271702345473397224319032, −3.38659604339457213573641019013, −1.495315901057767600133988057626, 1.63986741841392519145799806657, 2.67432953840826438432516584841, 4.35452780142998847721323942503, 5.65387172824109091840389573323, 6.24363896107647580741828263588, 7.06669283065434369110712116303, 9.19683037416593854339928486539, 10.265462101614692060034459255901, 11.50493356808370343784021070192, 12.00973170215230722303410499936, 13.22844620393294127592135632986, 13.9270822562452536232127687090, 15.082549558272557685947259207528, 16.3202591573096865793413927284, 17.04668861438984382247478756372, 18.57614373694505065039936633712, 18.92867871956607028630031381176, 20.47005399262798702235554497784, 21.67761819223500405513514508876, 22.231925543641308337384250349008, 22.645766496002840661554233907629, 24.19864085576940525121290056700, 24.610335297185792043337299353460, 25.51095746713658997954225955698, 26.933659669468337516581463234766

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