Properties

Degree 1
Conductor 199
Sign $0.865 - 0.501i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.857 − 0.513i)2-s + (0.296 − 0.954i)3-s + (0.472 + 0.881i)4-s + (0.723 + 0.690i)5-s + (−0.745 + 0.666i)6-s + (−0.386 + 0.922i)7-s + (0.0475 − 0.998i)8-s + (−0.823 − 0.567i)9-s + (−0.266 − 0.963i)10-s + (0.415 − 0.909i)11-s + (0.981 − 0.189i)12-s + (−0.0158 + 0.999i)13-s + (0.805 − 0.592i)14-s + (0.873 − 0.486i)15-s + (−0.553 + 0.832i)16-s + (0.580 + 0.814i)17-s + ⋯
L(s,χ)  = 1  + (−0.857 − 0.513i)2-s + (0.296 − 0.954i)3-s + (0.472 + 0.881i)4-s + (0.723 + 0.690i)5-s + (−0.745 + 0.666i)6-s + (−0.386 + 0.922i)7-s + (0.0475 − 0.998i)8-s + (−0.823 − 0.567i)9-s + (−0.266 − 0.963i)10-s + (0.415 − 0.909i)11-s + (0.981 − 0.189i)12-s + (−0.0158 + 0.999i)13-s + (0.805 − 0.592i)14-s + (0.873 − 0.486i)15-s + (−0.553 + 0.832i)16-s + (0.580 + 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.865 - 0.501i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.865 - 0.501i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(199\)
\( \varepsilon \)  =  $0.865 - 0.501i$
motivic weight  =  \(0\)
character  :  $\chi_{199} (53, \cdot )$
Sato-Tate  :  $\mu(99)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 199,\ (0:\ ),\ 0.865 - 0.501i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9229086737 - 0.2482651541i$
$L(\frac12,\chi)$  $\approx$  $0.9229086737 - 0.2482651541i$
$L(\chi,1)$  $\approx$  0.8583474250 - 0.2253384379i
$L(1,\chi)$  $\approx$  0.8583474250 - 0.2253384379i

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

−27.18213328300003888380900363862, −25.94871183908936916870083852803, −25.36127793578411097974751868299, −24.63737392839939980250707232482, −23.14468581321617548616566634422, −22.45124638146591770904009932839, −20.72527320591177368438202639571, −20.457055907124741189760782443513, −19.59950369525391545768852938665, −18.06877551275656428662445641323, −17.153936214936364993856384222796, −16.51632652546219486213068304343, −15.6794583983234940622715179282, −14.492775806713979408777949956673, −13.73649276919268358994642221763, −12.18744389419271503930266560216, −10.4871807094828806404780411742, −10.03463450247203263016904798020, −9.20547433060409708905138840546, −8.12909786872897206281557162597, −6.92515992740906156606237697912, −5.518245251790145736086795444804, −4.63459676366383185603309243892, −2.9116174353735668224720000253, −1.14133078729219785951815772973, 1.391057842746801144647805369387, 2.51609398314242028263670667624, 3.35704363735737657712786643943, 5.965061242267952911073338106432, 6.64721513308420354312781133121, 7.87360365522974826089785900573, 9.017876119893080438915236299055, 9.6578795872987224548268579123, 11.233881929940342517627633029323, 11.86709455600971703680196358185, 13.08579032517797163799132260615, 13.93804969395438533643502076122, 15.19170331814819204757472630904, 16.6122398084324009849847673992, 17.557578898630212592623962481620, 18.45027014767272015254068542348, 19.08815069266708901396414163136, 19.66401542838060605609617921367, 21.35134432171481309812065552057, 21.6489192162606231818418100959, 22.981173309254919738226974586261, 24.52979792386925827082661064118, 25.01145934462739663094054463859, 26.12146955156355301939169315867, 26.44939766732331413442936854277

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