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Properties

Label 1-571-571.418-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.594 - 0.803i$
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  + (−0.298 + 0.954i)2-s + (0.137 + 0.990i)3-s + (−0.821 − 0.569i)4-s + (0.0275 + 0.999i)5-s + (−0.986 − 0.164i)6-s + (−0.401 − 0.915i)7-s + (0.789 − 0.614i)8-s + (−0.962 + 0.272i)9-s + (−0.962 − 0.272i)10-s + (0.716 + 0.697i)11-s + (0.451 − 0.892i)12-s + (−0.191 + 0.981i)13-s + (0.993 − 0.110i)14-s + (−0.986 + 0.164i)15-s + (0.350 + 0.936i)16-s + (0.350 + 0.936i)17-s + ⋯
L(s)  = 1  + (−0.298 + 0.954i)2-s + (0.137 + 0.990i)3-s + (−0.821 − 0.569i)4-s + (0.0275 + 0.999i)5-s + (−0.986 − 0.164i)6-s + (−0.401 − 0.915i)7-s + (0.789 − 0.614i)8-s + (−0.962 + 0.272i)9-s + (−0.962 − 0.272i)10-s + (0.716 + 0.697i)11-s + (0.451 − 0.892i)12-s + (−0.191 + 0.981i)13-s + (0.993 − 0.110i)14-s + (−0.986 + 0.164i)15-s + (0.350 + 0.936i)16-s + (0.350 + 0.936i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.594 - 0.803i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.594 - 0.803i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3530114851 + 0.7004989339i\)
\(L(\frac12)\) \(\approx\) \(-0.3530114851 + 0.7004989339i\)
\(L(1)\) \(\approx\) \(0.3962602950 + 0.6779258933i\)
\(L(1)\) \(\approx\) \(0.3962602950 + 0.6779258933i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.298 + 0.954i)T \)
3 \( 1 + (0.137 + 0.990i)T \)
5 \( 1 + (0.0275 + 0.999i)T \)
7 \( 1 + (-0.401 - 0.915i)T \)
11 \( 1 + (0.716 + 0.697i)T \)
13 \( 1 + (-0.191 + 0.981i)T \)
17 \( 1 + (0.350 + 0.936i)T \)
19 \( 1 + (0.137 + 0.990i)T \)
23 \( 1 + (0.789 + 0.614i)T \)
29 \( 1 + (-0.754 - 0.656i)T \)
31 \( 1 + (-0.986 - 0.164i)T \)
37 \( 1 + (-0.191 - 0.981i)T \)
41 \( 1 + (0.0275 + 0.999i)T \)
43 \( 1 + (-0.962 - 0.272i)T \)
47 \( 1 + (0.635 - 0.771i)T \)
53 \( 1 + (-0.298 - 0.954i)T \)
59 \( 1 + (-0.401 - 0.915i)T \)
61 \( 1 + (0.975 - 0.218i)T \)
67 \( 1 + (0.975 + 0.218i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.754 + 0.656i)T \)
79 \( 1 + (0.904 + 0.426i)T \)
83 \( 1 + (0.635 - 0.771i)T \)
89 \( 1 + (0.350 + 0.936i)T \)
97 \( 1 + (-0.821 - 0.569i)T \)
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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

−22.451173412399616302596893560549, −21.95905141138602989181105011360, −20.70854346494718548815106033289, −20.16138068983645957332652095895, −19.37948661346634185773397663776, −18.69631857048767341919189244273, −17.9243957388309325018687120883, −17.06976828094789948357407475891, −16.30609852276392199787902836472, −14.94467447506330275059833890196, −13.76659684846064094476588434581, −13.09426917922936364991732826395, −12.39611523981059515046373818717, −11.83276225180495451179708790525, −10.90635685227229777680396757757, −9.34502267410300247852689472968, −8.98710186747642963356602105978, −8.19082535253863706475888962617, −7.10363651078591494037626410186, −5.70393326676729490497998002835, −4.979816114096730966111993802032, −3.35758728477243124301905407267, −2.64928857725103650769425803638, −1.42441114478847262722076333042, −0.464635921518785478358562377386, 1.8016404081847388221332309665, 3.77539211180391937484486522721, 3.89490652183307681427215238861, 5.31645531499682673418485959047, 6.37595831433283589624377851236, 7.11186265993469440568319208015, 8.00721517648772067788843055568, 9.33387868877673507897915225510, 9.817727673365863755982682380586, 10.57662758062001210798867982958, 11.51352830881432185378542732897, 13.08161819256680933323861829671, 14.20229182695462917412410853479, 14.57042699708315133836939556944, 15.26493502421995727131997938045, 16.3557604438771952555042317599, 16.93286673560070924102154123293, 17.57348440385389649034494495051, 18.87426787029837507369443108944, 19.424935747677316092298915621811, 20.37386475982331446459409768573, 21.617681600762355939559800580723, 22.224292391881174605023682886457, 23.19939862406299942909907057788, 23.36876581441314381480574346914

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