Properties

Degree 1
Conductor 199
Sign $0.939 + 0.342i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(199\)
\( \varepsilon \)  =  $0.939 + 0.342i$
motivic weight  =  \(0\)
character  :  $\chi_{199} (49, \cdot )$
Sato-Tate  :  $\mu(99)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 199,\ (0:\ ),\ 0.939 + 0.342i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.795837101 + 0.3171665210i$
$L(\frac12,\chi)$  $\approx$  $1.795837101 + 0.3171665210i$
$L(\chi,1)$  $\approx$  1.577401122 + 0.2397136703i
$L(1,\chi)$  $\approx$  1.577401122 + 0.2397136703i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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