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Properties

Label 1-571-571.9-r0-0-0
Degree $1$
Conductor $571$
Sign $0.0260 - 0.999i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
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  + (0.851 − 0.523i)2-s + (0.999 + 0.0220i)3-s + (0.451 − 0.892i)4-s + (−0.992 + 0.120i)5-s + (0.863 − 0.504i)6-s + (0.371 − 0.928i)7-s + (−0.0825 − 0.996i)8-s + (0.999 + 0.0440i)9-s + (−0.782 + 0.622i)10-s + (0.930 − 0.366i)11-s + (0.471 − 0.882i)12-s + (0.0935 + 0.995i)13-s + (−0.170 − 0.985i)14-s + (−0.995 + 0.0990i)15-s + (−0.592 − 0.805i)16-s + (−0.949 + 0.314i)17-s + ⋯
L(s)  = 1  + (0.851 − 0.523i)2-s + (0.999 + 0.0220i)3-s + (0.451 − 0.892i)4-s + (−0.992 + 0.120i)5-s + (0.863 − 0.504i)6-s + (0.371 − 0.928i)7-s + (−0.0825 − 0.996i)8-s + (0.999 + 0.0440i)9-s + (−0.782 + 0.622i)10-s + (0.930 − 0.366i)11-s + (0.471 − 0.882i)12-s + (0.0935 + 0.995i)13-s + (−0.170 − 0.985i)14-s + (−0.995 + 0.0990i)15-s + (−0.592 − 0.805i)16-s + (−0.949 + 0.314i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0260 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0260 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.0260 - 0.999i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.0260 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.082015024 - 2.028448272i\)
\(L(\frac12)\) \(\approx\) \(2.082015024 - 2.028448272i\)
\(L(1)\) \(\approx\) \(1.857219673 - 0.9697244959i\)
\(L(1)\) \(\approx\) \(1.857219673 - 0.9697244959i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.851 - 0.523i)T \)
3 \( 1 + (0.999 + 0.0220i)T \)
5 \( 1 + (-0.992 + 0.120i)T \)
7 \( 1 + (0.371 - 0.928i)T \)
11 \( 1 + (0.930 - 0.366i)T \)
13 \( 1 + (0.0935 + 0.995i)T \)
17 \( 1 + (-0.949 + 0.314i)T \)
19 \( 1 + (-0.795 - 0.605i)T \)
23 \( 1 + (0.652 - 0.757i)T \)
29 \( 1 + (-0.298 - 0.954i)T \)
31 \( 1 + (-0.401 + 0.915i)T \)
37 \( 1 + (0.509 + 0.860i)T \)
41 \( 1 + (-0.191 + 0.981i)T \)
43 \( 1 + (-0.834 - 0.551i)T \)
47 \( 1 + (0.993 - 0.110i)T \)
53 \( 1 + (0.761 - 0.648i)T \)
59 \( 1 + (0.245 + 0.969i)T \)
61 \( 1 + (0.565 - 0.824i)T \)
67 \( 1 + (-0.942 - 0.335i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.999 + 0.0110i)T \)
79 \( 1 + (0.775 + 0.631i)T \)
83 \( 1 + (-0.868 - 0.495i)T \)
89 \( 1 + (0.00551 + 0.999i)T \)
97 \( 1 + (0.988 + 0.153i)T \)
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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

−23.60266731815797270661943897546, −22.56441122420770013012264333333, −21.94719472698568834943172264191, −20.92749373602416380223957687118, −20.2218741621797273699521262816, −19.567711662577048809751717568726, −18.52630031218011620566916514920, −17.53042344225408564303532661543, −16.365797702707114103500526291796, −15.44724663464339585189096068521, −15.026333106759885543178812563261, −14.48254094019298674810133597290, −13.22190845101944173590879913437, −12.58362165294255891087105091717, −11.78994130236651559362389277761, −10.83466251340539158928648400584, −9.11142760845593858267266569354, −8.59361584565481856524638222964, −7.67198349670697473069542096965, −6.98180402923091323166002426743, −5.71057828166306167566150242160, −4.57083343967589476602272855022, −3.82507015668055937599434717269, −2.91491349266747665064757098958, −1.83127045550601766657511992585, 1.10173215777956393407628711310, 2.25075736037673998583483051771, 3.42733864702082124172518167339, 4.2265690945486939373161350175, 4.56561027583717381381698145881, 6.66871501814328025351633798928, 6.966943344966225070732746420561, 8.34556938496705783366237094610, 9.1207260611104708337793184206, 10.37963338289155726928482344143, 11.163945946674391936394703022303, 11.88219802054483576593943141358, 13.079043447245173802171137878928, 13.66719808394185478907564767103, 14.64654999132536007488095202344, 14.98468651661523963385395565840, 16.05896902847241612508182196297, 16.94587410689022291478103738450, 18.541176831397825351113203027683, 19.29906026799911264715687944400, 19.82265873422222524418088300687, 20.41429772021663553518695769627, 21.356228566843549510561992190911, 22.06764346788269638823072950985, 23.13121530831034014179305849264

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