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Properties

Label 1-569-569.126-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.272 + 0.962i$
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  + (0.952 + 0.304i)2-s + (−0.262 + 0.964i)3-s + (0.814 + 0.580i)4-s + (−0.999 − 0.0442i)5-s + (−0.544 + 0.839i)6-s + (0.730 − 0.683i)7-s + (0.598 + 0.801i)8-s + (−0.862 − 0.506i)9-s + (−0.937 − 0.346i)10-s + (0.850 + 0.525i)11-s + (−0.773 + 0.633i)12-s + (0.787 + 0.616i)13-s + (0.903 − 0.428i)14-s + (0.304 − 0.952i)15-s + (0.325 + 0.945i)16-s + (0.110 − 0.993i)17-s + ⋯
L(s)  = 1  + (0.952 + 0.304i)2-s + (−0.262 + 0.964i)3-s + (0.814 + 0.580i)4-s + (−0.999 − 0.0442i)5-s + (−0.544 + 0.839i)6-s + (0.730 − 0.683i)7-s + (0.598 + 0.801i)8-s + (−0.862 − 0.506i)9-s + (−0.937 − 0.346i)10-s + (0.850 + 0.525i)11-s + (−0.773 + 0.633i)12-s + (0.787 + 0.616i)13-s + (0.903 − 0.428i)14-s + (0.304 − 0.952i)15-s + (0.325 + 0.945i)16-s + (0.110 − 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.272 + 0.962i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.272 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.314513287 + 1.739372999i\)
\(L(\frac12)\) \(\approx\) \(1.314513287 + 1.739372999i\)
\(L(1)\) \(\approx\) \(1.396030668 + 0.8724076292i\)
\(L(1)\) \(\approx\) \(1.396030668 + 0.8724076292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.952 + 0.304i)T \)
3 \( 1 + (-0.262 + 0.964i)T \)
5 \( 1 + (-0.999 - 0.0442i)T \)
7 \( 1 + (0.730 - 0.683i)T \)
11 \( 1 + (0.850 + 0.525i)T \)
13 \( 1 + (0.787 + 0.616i)T \)
17 \( 1 + (0.110 - 0.993i)T \)
19 \( 1 + (0.580 + 0.814i)T \)
23 \( 1 + (-0.467 + 0.883i)T \)
29 \( 1 + (-0.993 + 0.110i)T \)
31 \( 1 + (-0.958 + 0.283i)T \)
37 \( 1 + (0.683 + 0.730i)T \)
41 \( 1 + (-0.883 - 0.467i)T \)
43 \( 1 + (-0.952 + 0.304i)T \)
47 \( 1 + (0.745 + 0.666i)T \)
53 \( 1 + (0.544 - 0.839i)T \)
59 \( 1 + (-0.745 - 0.666i)T \)
61 \( 1 + (0.991 + 0.132i)T \)
67 \( 1 + (0.110 + 0.993i)T \)
71 \( 1 + (0.598 - 0.801i)T \)
73 \( 1 + (-0.580 - 0.814i)T \)
79 \( 1 + (-0.408 + 0.912i)T \)
83 \( 1 + (-0.997 - 0.0663i)T \)
89 \( 1 + (0.958 + 0.283i)T \)
97 \( 1 + (0.0883 - 0.996i)T \)
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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

−23.1216522057627514425431681538, −22.23143937994554612890600893847, −21.66702202801572017059420149668, −20.15759854647672245941040674731, −20.021057657084411211167725737, −18.76409899751581286763575186668, −18.44170452340284821617743466015, −17.06726208986163576220538968999, −16.15462779380651688822600272016, −15.09697978698241401637125595046, −14.57487932339787829157275714709, −13.51803196928941206759142821241, −12.69182073020971333407988191083, −11.92088910523322278657835158282, −11.327313268819812409762952499204, −10.73221189274836292698824569839, −8.84030888242270453907295574792, −8.03666944160478865813678085194, −7.0920069511596073681571996152, −6.08298103345717118314655461989, −5.3917227113886580978186228436, −4.14065120068655560595479342562, −3.20786042521865878346695817359, −2.018384966612376043723060250, −0.94551225836236355002429596814, 1.54603233417405590963362709081, 3.42261742825922900464977068917, 3.892671323644477949480832349792, 4.664396977870589028373071878156, 5.53575213145144754011703521511, 6.81128041513642872232412817411, 7.60207531689106612281167358643, 8.61266305273779832687085820375, 9.80429400826873318549151328725, 11.09496863327210823397617335478, 11.510243697551986962687803379536, 12.13959690494089301889105173439, 13.58731026909798080595430282559, 14.41320934876817808823356893365, 14.96246414928351089899829699224, 15.9513184528072609697845466049, 16.47469234678630069601087689213, 17.20927638785710214184331806772, 18.39833056549417680487827352216, 19.936898230373312412242120897199, 20.36031669583832132480160751419, 20.981943252688449484132225133971, 22.074451748611469784528187074024, 22.682625540110670525759892425561, 23.47500227002077823345748483666

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