Properties

Degree 1
Conductor 191
Sign $0.957 - 0.287i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.180 + 0.983i)2-s + (−0.461 − 0.887i)3-s + (−0.934 + 0.355i)4-s + (0.546 − 0.837i)5-s + (0.789 − 0.614i)6-s + (0.309 + 0.951i)7-s + (−0.518 − 0.854i)8-s + (−0.574 + 0.818i)9-s + (0.922 + 0.386i)10-s + (−0.677 − 0.735i)11-s + (0.746 + 0.665i)12-s + (0.894 − 0.446i)13-s + (−0.879 + 0.475i)14-s + (−0.995 − 0.0990i)15-s + (0.746 − 0.665i)16-s + (0.371 − 0.928i)17-s + ⋯
L(s,χ)  = 1  + (0.180 + 0.983i)2-s + (−0.461 − 0.887i)3-s + (−0.934 + 0.355i)4-s + (0.546 − 0.837i)5-s + (0.789 − 0.614i)6-s + (0.309 + 0.951i)7-s + (−0.518 − 0.854i)8-s + (−0.574 + 0.818i)9-s + (0.922 + 0.386i)10-s + (−0.677 − 0.735i)11-s + (0.746 + 0.665i)12-s + (0.894 − 0.446i)13-s + (−0.879 + 0.475i)14-s + (−0.995 − 0.0990i)15-s + (0.746 − 0.665i)16-s + (0.371 − 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.957 - 0.287i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.957 - 0.287i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(191\)
\( \varepsilon \)  =  $0.957 - 0.287i$
motivic weight  =  \(0\)
character  :  $\chi_{191} (46, \cdot )$
Sato-Tate  :  $\mu(95)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 191,\ (0:\ ),\ 0.957 - 0.287i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.031194092 - 0.1513902700i$
$L(\frac12,\chi)$  $\approx$  $1.031194092 - 0.1513902700i$
$L(\chi,1)$  $\approx$  0.9951494103 + 0.04864400897i
$L(1,\chi)$  $\approx$  0.9951494103 + 0.04864400897i

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

−27.106682242542369708848826579390, −26.48315417730020171510379726662, −25.7127527170854172933420433179, −23.562143961413710441038351534634, −23.23137140361585360129417062849, −22.18683265209489794871340688514, −21.3426368265570268725998579649, −20.69771167990857545546673030531, −19.74664107475337539436196036207, −18.23179759691960628226314155257, −17.80074093277540362903297780207, −16.617372423196824513413252061250, −15.22772799460669618278918042500, −14.24788987619549924629443104177, −13.48379742990210577197668470071, −12.0241058786097702186872475641, −11.022482483626366774808170403883, −10.23990947765546296415345972908, −9.797587308269634728258495954003, −8.179203147463971191469606721902, −6.44929787305574043553208347881, −5.2670718145264029902377736594, −4.11006233161812422276751233074, −3.19383647886835583441325906990, −1.55817629428616381544696188148, 0.92795349250675786889097961025, 2.77269512648983091706112101707, 4.91296918558600747008361354189, 5.62435961029131815424122814927, 6.36103849388225606103063849264, 7.99765482891678226175594802562, 8.423832844744754947279742223317, 9.75186806634563099788098376263, 11.55419412013561163806794095071, 12.47287782286980402022538938693, 13.48297717315862975741650568157, 13.983826730616606502136370455440, 15.71659250282948578051728222082, 16.24642736456233189031613158903, 17.463929032889143580882165794060, 18.17688773977222221176468903294, 18.79256464928640998874953934850, 20.49037274214439853791438320437, 21.59131001955351544113460315968, 22.471341902306218006330511945610, 23.605214863429218883489292475304, 24.26565874096837326283727010615, 25.03373427920025953874516684868, 25.55938050536737888891563558097, 26.976541066294556533411176970220

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