Properties

Degree 1
Conductor 229
Sign $0.427 - 0.904i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.475 + 0.879i)2-s + (0.245 + 0.969i)3-s + (−0.546 − 0.837i)4-s + (0.677 − 0.735i)5-s + (−0.969 − 0.245i)6-s + (−0.164 − 0.986i)7-s + (0.996 − 0.0825i)8-s + (−0.879 + 0.475i)9-s + (0.324 + 0.945i)10-s + (−0.789 + 0.614i)11-s + (0.677 − 0.735i)12-s + (0.735 + 0.677i)13-s + (0.945 + 0.324i)14-s + (0.879 + 0.475i)15-s + (−0.401 + 0.915i)16-s + (−0.677 + 0.735i)17-s + ⋯
L(s,χ)  = 1  + (−0.475 + 0.879i)2-s + (0.245 + 0.969i)3-s + (−0.546 − 0.837i)4-s + (0.677 − 0.735i)5-s + (−0.969 − 0.245i)6-s + (−0.164 − 0.986i)7-s + (0.996 − 0.0825i)8-s + (−0.879 + 0.475i)9-s + (0.324 + 0.945i)10-s + (−0.789 + 0.614i)11-s + (0.677 − 0.735i)12-s + (0.735 + 0.677i)13-s + (0.945 + 0.324i)14-s + (0.879 + 0.475i)15-s + (−0.401 + 0.915i)16-s + (−0.677 + 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.427 - 0.904i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.427 - 0.904i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $0.427 - 0.904i$
motivic weight  =  \(0\)
character  :  $\chi_{229} (136, \cdot )$
Sato-Tate  :  $\mu(76)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 229,\ (1:\ ),\ 0.427 - 0.904i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4948270541 - 0.3135581287i$
$L(\frac12,\chi)$  $\approx$  $0.4948270541 - 0.3135581287i$
$L(\chi,1)$  $\approx$  0.7066798898 + 0.2645872301i
$L(1,\chi)$  $\approx$  0.7066798898 + 0.2645872301i

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

−26.188089863127242458139027500708, −25.45796571800940897205139161984, −24.84828716086695041333894502449, −23.28578868833738989661314682441, −22.51392573997385269997888298689, −21.441055332865278206451199872161, −20.751045288720846967119983932443, −19.44931909220407682095382556380, −18.729854340531934238244463922697, −18.13180233203180547833732231882, −17.50600725601115039264577597422, −15.96193634500031317821998687678, −14.61734507657324307416827132038, −13.36676344255090412630509309780, −13.0498556334273019774667737161, −11.7154309687780251994104131087, −10.94236175115892068036468681472, −9.73109492659626741442207360781, −8.66767487876023869200536965633, −7.85502580318721888363442651726, −6.47897495045027024744695465056, −5.45014961620524569635867651842, −3.219565603437733023461704269002, −2.61330169428276739624995597104, −1.49126218169098331042600098065, 0.21059574077104166946942988205, 1.965615432498625887191471724295, 4.11372746998531053950859987790, 4.797503414933044363615072610693, 5.979093584603598254852349900867, 7.17467195702759458099158148028, 8.60212851798568668225936718674, 9.09050397620808281590383081065, 10.33652283561948377831370056692, 10.75185061649791996596013162802, 12.98423460921467331503503137006, 13.70008624171128864681945014231, 14.72046597473901797534289949149, 15.72636888638863558623353461597, 16.561011066942753622333715960873, 17.14664604952549415501581788592, 18.11315349380216078220778321202, 19.55899209287179667506203807538, 20.36171170956453427156303430755, 21.202175089847914875701147410164, 22.340576674802999981694519340940, 23.44129867398260335674266611798, 24.10171366645653471596504174652, 25.41750134399666658104360129822, 26.0678609549061778627513837479

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