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Properties

Label 1-179-179.27-r0-0-0
Degree $1$
Conductor $179$
Sign $0.907 + 0.420i$
Analytic cond. $0.831272$
Root an. cond. $0.831272$
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  + (0.427 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.907 + 0.420i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.907 + 0.420i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7689939614 + 0.1695938870i\)
\(L(\frac12)\) \(\approx\) \(0.7689939614 + 0.1695938870i\)
\(L(1)\) \(\approx\) \(0.8423672401 - 0.1539200153i\)
\(L(1)\) \(\approx\) \(0.8423672401 - 0.1539200153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.427 - 0.904i)T \)
3 \( 1 + (-0.863 - 0.505i)T \)
5 \( 1 + (0.362 + 0.932i)T \)
7 \( 1 + (-0.0529 + 0.998i)T \)
11 \( 1 + (-0.329 + 0.944i)T \)
13 \( 1 + (-0.999 - 0.0352i)T \)
17 \( 1 + (0.550 + 0.835i)T \)
19 \( 1 + (-0.261 + 0.965i)T \)
23 \( 1 + (-0.123 - 0.992i)T \)
29 \( 1 + (0.227 - 0.973i)T \)
31 \( 1 + (0.607 - 0.794i)T \)
37 \( 1 + (0.227 + 0.973i)T \)
41 \( 1 + (0.662 + 0.749i)T \)
43 \( 1 + (-0.737 - 0.675i)T \)
47 \( 1 + (-0.984 - 0.175i)T \)
53 \( 1 + (0.0176 + 0.999i)T \)
59 \( 1 + (0.0881 + 0.996i)T \)
61 \( 1 + (0.938 + 0.345i)T \)
67 \( 1 + (-0.969 - 0.244i)T \)
71 \( 1 + (-0.783 + 0.621i)T \)
73 \( 1 + (0.911 + 0.411i)T \)
79 \( 1 + (0.990 + 0.140i)T \)
83 \( 1 + (-0.394 - 0.918i)T \)
89 \( 1 + (0.427 + 0.904i)T \)
97 \( 1 + (0.960 - 0.278i)T \)
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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

−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