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Properties

Label 1-569-569.164-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.784 + 0.619i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.784 + 0.619i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.784 + 0.619i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4292310644 + 1.236510989i\)
\(L(\frac12)\) \(\approx\) \(0.4292310644 + 1.236510989i\)
\(L(1)\) \(\approx\) \(0.7848354255 + 0.6824125689i\)
\(L(1)\) \(\approx\) \(0.7848354255 + 0.6824125689i\)

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

−23.08717527928142335492547533469, −21.44062701371703056673417730927, −21.19580883440344674583221596956, −20.17882217680990947322809905672, −19.585182374246255097371483015878, −19.249006998134339929211054731292, −18.01741795160142048966362820706, −17.18954519400715716998854702079, −16.22040273000266435794557587587, −15.6326281461349065217984380812, −14.16449549275818904320947632311, −13.26995692911520918686811854133, −12.93334626372003352480068267974, −11.6856896613002183032245359059, −10.85685631093890889606436196293, −9.86448695919058801444092180373, −8.96195602507322752387629296371, −8.15849073772632258391297109010, −7.77574813371253983410508853711, −6.48759768044596006657871094583, −4.66574301787529934766097267388, −3.66189110789442888918906405690, −3.2207881701389725811619581674, −1.54272092387296190879466415501, −0.8348022958018751424107588697, 1.67354571562786391695954940070, 2.6403178267280629602784764450, 3.86779872408599589070703953756, 4.99090559285354831438471919507, 6.34479038190753155707231528182, 7.059946425598460686070108871968, 8.070928925704848078823588019976, 8.626525554827570718441282341648, 9.690637598980310251201742779776, 10.26723733579934428064757342936, 11.47710738985749900101145010372, 12.5161164341755983523875043397, 13.96904407354851088496345634865, 14.462159673235283040412081915343, 15.28574239885089201482776379420, 15.750388887172656125427946321126, 16.66335238372533097681886021491, 18.150131613373917786581217636406, 18.599904429807908690286706250039, 19.04117386744953395891316606434, 20.17992611968692106435120331606, 20.80083692633223796331731048020, 22.102186974597776985019625922241, 22.848228698250358969353820996526, 23.70979509372629797155780361132

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