Properties

Degree 1
Conductor 179
Sign $0.907 + 0.420i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.427 − 0.904i)2-s + (−0.863 − 0.505i)3-s + (−0.635 − 0.772i)4-s + (0.362 + 0.932i)5-s + (−0.825 + 0.564i)6-s + (−0.0529 + 0.998i)7-s + (−0.969 + 0.244i)8-s + (0.489 + 0.871i)9-s + (0.997 + 0.0705i)10-s + (−0.329 + 0.944i)11-s + (0.158 + 0.987i)12-s + (−0.999 − 0.0352i)13-s + (0.880 + 0.474i)14-s + (0.158 − 0.987i)15-s + (−0.192 + 0.981i)16-s + (0.550 + 0.835i)17-s + ⋯
L(s,χ)  = 1  + (0.427 − 0.904i)2-s + (−0.863 − 0.505i)3-s + (−0.635 − 0.772i)4-s + (0.362 + 0.932i)5-s + (−0.825 + 0.564i)6-s + (−0.0529 + 0.998i)7-s + (−0.969 + 0.244i)8-s + (0.489 + 0.871i)9-s + (0.997 + 0.0705i)10-s + (−0.329 + 0.944i)11-s + (0.158 + 0.987i)12-s + (−0.999 − 0.0352i)13-s + (0.880 + 0.474i)14-s + (0.158 − 0.987i)15-s + (−0.192 + 0.981i)16-s + (0.550 + 0.835i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.907 + 0.420i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.907 + 0.420i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(179\)
\( \varepsilon \)  =  $0.907 + 0.420i$
motivic weight  =  \(0\)
character  :  $\chi_{179} (27, \cdot )$
Sato-Tate  :  $\mu(89)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 179,\ (0:\ ),\ 0.907 + 0.420i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7689939614 + 0.1695938870i$
$L(\frac12,\chi)$  $\approx$  $0.7689939614 + 0.1695938870i$
$L(\chi,1)$  $\approx$  0.8423672401 - 0.1539200153i
$L(1,\chi)$  $\approx$  0.8423672401 - 0.1539200153i

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

−27.10071268628028986378775836166, −26.46760552286887679201317890189, −25.217543144048688092474158452374, −24.053753971826687785938410721101, −23.71219849680040574597895166442, −22.63127179733621219082963670863, −21.571901378029195153232231081412, −21.03395789905217861073449880480, −19.64326013173733576774784851654, −17.87213201438254903159243776348, −17.29032583868301637960928827371, −16.347411598400588075693793155107, −15.971670925249996818024515525691, −14.44916785486634752216986582595, −13.458416062134835350102666309844, −12.55026556530199979766683978990, −11.41114689288751400136870000101, −9.98482680533613256163066951552, −9.05313138989193028171620082059, −7.63729961631179202511388211425, −6.54373457089439705608225108508, −5.2596243898288643325340784683, −4.77567148840597140460118980751, −3.42834177421327558301220711115, −0.61038463488103538313463977528, 1.876885780230801100246913080694, 2.66199227427775783233504388568, 4.487600915500986413766731982557, 5.67956757681538756476171503874, 6.4463601690573754709974722512, 7.97799269838077755850196490763, 9.88419345331615843299110477948, 10.317728916860507261474924129724, 11.681428039511445820420024633941, 12.29494740218591158757892318440, 13.18536204540374328959473285015, 14.583521527821857385401123472547, 15.21702187167611168910928335015, 17.02561258179566614631628524011, 18.02534176990962814344099676959, 18.7002519758930110292533750496, 19.392854323469973523995273266771, 20.96111953088632725923068176049, 21.83301133579036290213725865811, 22.55285845471250069140524327785, 23.16015960120744474157077636099, 24.37423396840440231811325459458, 25.31710110850367500418903002315, 26.70110857319806315799845705103, 27.85013268993102927504615754603

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