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Properties

Label 1-229-229.32-r1-0-0
Degree $1$
Conductor $229$
Sign $0.427 + 0.904i$
Analytic cond. $24.6094$
Root an. cond. $24.6094$
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  + (−0.475 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.427 + 0.904i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ 0.427 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4948270541 + 0.3135581287i\)
\(L(\frac12)\) \(\approx\) \(0.4948270541 + 0.3135581287i\)
\(L(1)\) \(\approx\) \(0.7066798898 - 0.2645872301i\)
\(L(1)\) \(\approx\) \(0.7066798898 - 0.2645872301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (-0.475 - 0.879i)T \)
3 \( 1 + (0.245 - 0.969i)T \)
5 \( 1 + (0.677 + 0.735i)T \)
7 \( 1 + (-0.164 + 0.986i)T \)
11 \( 1 + (-0.789 - 0.614i)T \)
13 \( 1 + (0.735 - 0.677i)T \)
17 \( 1 + (-0.677 - 0.735i)T \)
19 \( 1 + (-0.677 + 0.735i)T \)
23 \( 1 + (0.324 + 0.945i)T \)
29 \( 1 + (0.164 - 0.986i)T \)
31 \( 1 + (-0.614 + 0.789i)T \)
37 \( 1 + (-0.401 + 0.915i)T \)
41 \( 1 + (0.475 + 0.879i)T \)
43 \( 1 + (-0.401 + 0.915i)T \)
47 \( 1 + (-0.475 + 0.879i)T \)
53 \( 1 + (0.245 - 0.969i)T \)
59 \( 1 + (0.915 - 0.401i)T \)
61 \( 1 + (-0.879 - 0.475i)T \)
67 \( 1 + (-0.475 + 0.879i)T \)
71 \( 1 + (-0.789 + 0.614i)T \)
73 \( 1 + (-0.996 - 0.0825i)T \)
79 \( 1 + (0.164 + 0.986i)T \)
83 \( 1 + (-0.401 - 0.915i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.0825 + 0.996i)T \)
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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

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

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