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Properties

Label 1-569-569.170-r0-0-0
Degree $1$
Conductor $569$
Sign $0.713 - 0.700i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (−0.666 − 0.745i)2-s + (−0.984 + 0.176i)3-s + (−0.110 + 0.993i)4-s + (−0.525 − 0.850i)5-s + (0.787 + 0.616i)6-s + (−0.0221 − 0.999i)7-s + (0.814 − 0.580i)8-s + (0.937 − 0.346i)9-s + (−0.283 + 0.958i)10-s + (−0.154 + 0.988i)11-s + (−0.0663 − 0.997i)12-s + (0.903 − 0.428i)13-s + (−0.730 + 0.683i)14-s + (0.666 + 0.745i)15-s + (−0.975 − 0.219i)16-s + (0.562 + 0.826i)17-s + ⋯
L(s)  = 1  + (−0.666 − 0.745i)2-s + (−0.984 + 0.176i)3-s + (−0.110 + 0.993i)4-s + (−0.525 − 0.850i)5-s + (0.787 + 0.616i)6-s + (−0.0221 − 0.999i)7-s + (0.814 − 0.580i)8-s + (0.937 − 0.346i)9-s + (−0.283 + 0.958i)10-s + (−0.154 + 0.988i)11-s + (−0.0663 − 0.997i)12-s + (0.903 − 0.428i)13-s + (−0.730 + 0.683i)14-s + (0.666 + 0.745i)15-s + (−0.975 − 0.219i)16-s + (0.562 + 0.826i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5521482399 - 0.2256893945i\)
\(L(\frac12)\) \(\approx\) \(0.5521482399 - 0.2256893945i\)
\(L(1)\) \(\approx\) \(0.5235380259 - 0.1973566621i\)
\(L(1)\) \(\approx\) \(0.5235380259 - 0.1973566621i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.666 - 0.745i)T \)
3 \( 1 + (-0.984 + 0.176i)T \)
5 \( 1 + (-0.525 - 0.850i)T \)
7 \( 1 + (-0.0221 - 0.999i)T \)
11 \( 1 + (-0.154 + 0.988i)T \)
13 \( 1 + (0.903 - 0.428i)T \)
17 \( 1 + (0.562 + 0.826i)T \)
19 \( 1 + (0.110 + 0.993i)T \)
23 \( 1 + (0.197 - 0.980i)T \)
29 \( 1 + (-0.562 + 0.826i)T \)
31 \( 1 + (-0.325 - 0.945i)T \)
37 \( 1 + (0.0221 + 0.999i)T \)
41 \( 1 + (-0.197 + 0.980i)T \)
43 \( 1 + (-0.666 + 0.745i)T \)
47 \( 1 + (0.883 - 0.467i)T \)
53 \( 1 + (0.787 + 0.616i)T \)
59 \( 1 + (0.883 - 0.467i)T \)
61 \( 1 + (0.996 - 0.0883i)T \)
67 \( 1 + (0.562 - 0.826i)T \)
71 \( 1 + (0.814 + 0.580i)T \)
73 \( 1 + (0.110 + 0.993i)T \)
79 \( 1 + (0.240 - 0.970i)T \)
83 \( 1 + (0.999 - 0.0442i)T \)
89 \( 1 + (-0.325 + 0.945i)T \)
97 \( 1 + (0.448 - 0.894i)T \)
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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

−23.555063403454948817957023942586, −22.69921874168382211652514430866, −21.97889637615114514195761609301, −21.06934544492170127834814893719, −19.41735912075454788545938963884, −18.91861155410961801411242527461, −18.28934050838415632738819046976, −17.695522123432375180098469597286, −16.5207608343504735715386619818, −15.80485103117852878120734249301, −15.41738693100247503338092161183, −14.17079068426543403291597776237, −13.33628890474573417963263467309, −11.81172710445386930526649204494, −11.31568227855687760246325476200, −10.59327931985656447663112543099, −9.42542787653616957881738755342, −8.52707509539028909990595676410, −7.42311946197715948565740846950, −6.741188251358121058861791780346, −5.78696370447528591569202309927, −5.2439823118573590673269574890, −3.69259525670878520000905907184, −2.20992586691451878127258503338, −0.68297160711406156592746580613, 0.84940051789677155479043926565, 1.61641406958704488985871831786, 3.637475552019523315338483139162, 4.17004597698955016877709621247, 5.18792793447627703569839070853, 6.5878043963709907543383690358, 7.643305026052176790018875368843, 8.32678254225226653666487976252, 9.651860464148236573038155115075, 10.30629824045582927909987260960, 11.046184965731784330772170478188, 11.96041237768164221596908653748, 12.75469866649607456372479584321, 13.19997831409350690561922840699, 14.905999785309962645890705301137, 16.04942206736641776906790019684, 16.75007063894283707323320638229, 17.11482868471928980418763008511, 18.18998578894777124467743451039, 18.84064200979104702454754912892, 20.16905133929390626552314214825, 20.45995371904980999472466403594, 21.21838712757696363107176957373, 22.39323804888446039058068315807, 23.15059298189040153400007813125

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