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Properties

Label 1-569-569.121-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.943 - 0.330i$
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 + (−0.176 + 0.984i)3-s + (−0.110 − 0.993i)4-s + (−0.525 + 0.850i)5-s + (0.616 + 0.787i)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 + (0.988 − 0.154i)11-s + (0.997 + 0.0663i)12-s + (−0.903 − 0.428i)13-s + (−0.730 − 0.683i)14-s + (−0.745 − 0.666i)15-s + (−0.975 + 0.219i)16-s + (−0.562 + 0.826i)17-s + ⋯
L(s)  = 1  + (0.666 − 0.745i)2-s + (−0.176 + 0.984i)3-s + (−0.110 − 0.993i)4-s + (−0.525 + 0.850i)5-s + (0.616 + 0.787i)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 + (0.988 − 0.154i)11-s + (0.997 + 0.0663i)12-s + (−0.903 − 0.428i)13-s + (−0.730 − 0.683i)14-s + (−0.745 − 0.666i)15-s + (−0.975 + 0.219i)16-s + (−0.562 + 0.826i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.943 - 0.330i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.943 - 0.330i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09106751797 - 0.5363643814i\)
\(L(\frac12)\) \(\approx\) \(0.09106751797 - 0.5363643814i\)
\(L(1)\) \(\approx\) \(0.8746320945 - 0.2668173883i\)
\(L(1)\) \(\approx\) \(0.8746320945 - 0.2668173883i\)

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

−23.78772859547160692788557133819, −23.02133728081227239779137601135, −22.09748167603784548095949447366, −21.49925873551591261047901380826, −20.09960679289205851958829380784, −19.6254090091662565626552447387, −18.41231055218178626496874637688, −17.69271340557886729366391740864, −16.71427098700500031688694724934, −16.244915563722598107127879420649, −15.00174628579564841041508763141, −14.43220618610250800007406061082, −13.34298627766177711836635859388, −12.51270471683564591095488079350, −11.99935405556966892600189179150, −11.45399878556110136261059781363, −9.29658853723347204207249294630, −8.64757299692938709198880826324, −7.83417676395330581806313566728, −6.83935497886839378679998289095, −6.120404771786832736454530346388, −5.05177321979782665109004561574, −4.319843803462635321830899164917, −2.84431528693533746281221500784, −1.76522960336761941063554341409, 0.21365660640570787960955563230, 2.100511380140462249883496975100, 3.343716766837369288041399850922, 4.01822052489723269294219674682, 4.596866581995647545729673853856, 6.02673927078415767362497434367, 6.74362777579000188294777051464, 8.14485332509802128603562070144, 9.469884604531447926201820064162, 10.271641200171493373559883274601, 10.83917425308605071298622388330, 11.55651276695750267086571792233, 12.45915345144525133585079077746, 13.71693359793810701362704460651, 14.568053488823475467751535750473, 14.95175770247676679064427820687, 15.91363063225439794397254306065, 17.05576807714775419675428113940, 17.76876049719648576716633773932, 19.21853727482203799549626638461, 19.76176889655212858736604747740, 20.293285205222660925665616202150, 21.487504666512495418144061902888, 22.10420461310278001471663398563, 22.57930564082870450955818265773

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