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Properties

Label 1-571-571.3-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.881 - 0.472i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (0.962 − 0.272i)2-s + (0.999 + 0.0110i)3-s + (0.851 − 0.523i)4-s + (−0.0605 − 0.998i)5-s + (0.965 − 0.261i)6-s + (−0.828 + 0.560i)7-s + (0.677 − 0.735i)8-s + (0.999 + 0.0220i)9-s + (−0.329 − 0.944i)10-s + (−0.982 + 0.186i)11-s + (0.857 − 0.514i)12-s + (−0.739 − 0.673i)13-s + (−0.644 + 0.764i)14-s + (−0.0495 − 0.998i)15-s + (0.451 − 0.892i)16-s + (−0.159 − 0.987i)17-s + ⋯
L(s)  = 1  + (0.962 − 0.272i)2-s + (0.999 + 0.0110i)3-s + (0.851 − 0.523i)4-s + (−0.0605 − 0.998i)5-s + (0.965 − 0.261i)6-s + (−0.828 + 0.560i)7-s + (0.677 − 0.735i)8-s + (0.999 + 0.0220i)9-s + (−0.329 − 0.944i)10-s + (−0.982 + 0.186i)11-s + (0.857 − 0.514i)12-s + (−0.739 − 0.673i)13-s + (−0.644 + 0.764i)14-s + (−0.0495 − 0.998i)15-s + (0.451 − 0.892i)16-s + (−0.159 − 0.987i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.881 - 0.472i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.881 - 0.472i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8512560317 - 3.391383742i\)
\(L(\frac12)\) \(\approx\) \(0.8512560317 - 3.391383742i\)
\(L(1)\) \(\approx\) \(1.804328165 - 1.027339911i\)
\(L(1)\) \(\approx\) \(1.804328165 - 1.027339911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.962 - 0.272i)T \)
3 \( 1 + (0.999 + 0.0110i)T \)
5 \( 1 + (-0.0605 - 0.998i)T \)
7 \( 1 + (-0.828 + 0.560i)T \)
11 \( 1 + (-0.982 + 0.186i)T \)
13 \( 1 + (-0.739 - 0.673i)T \)
17 \( 1 + (-0.159 - 0.987i)T \)
19 \( 1 + (0.319 - 0.947i)T \)
23 \( 1 + (-0.909 + 0.416i)T \)
29 \( 1 + (-0.592 + 0.805i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (-0.868 - 0.495i)T \)
41 \( 1 + (-0.635 - 0.771i)T \)
43 \( 1 + (0.287 - 0.957i)T \)
47 \( 1 + (0.998 - 0.0550i)T \)
53 \( 1 + (-0.938 + 0.345i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (0.884 - 0.466i)T \)
67 \( 1 + (0.170 - 0.985i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.00551 - 0.999i)T \)
79 \( 1 + (0.942 + 0.335i)T \)
83 \( 1 + (-0.256 + 0.966i)T \)
89 \( 1 + (0.709 + 0.705i)T \)
97 \( 1 + (-0.997 - 0.0770i)T \)
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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

−23.468467971524376965304083468142, −22.44129666488227610640880342907, −21.894406057490265950707192458966, −20.94577999795996590329620994347, −20.24098225417923178735406149276, −19.21116492437733240623692460925, −18.81450191488587517171858713909, −17.38859910010685296000277096898, −16.28825018168527976748446853127, −15.60210277838530804113686937260, −14.76085050548869955722451403794, −14.137169498612873098897120851983, −13.381479669334554903046722224023, −12.6763717429256365460555147190, −11.565972491916483079059314344055, −10.300764118668807807309300826793, −9.91845088021203859311444562119, −8.19814414679505038148017975414, −7.575877067169714117094320977153, −6.71986728566657816303468946598, −5.90949125102196483944098958408, −4.32961982705090278146571899509, −3.657121920438508186952049028, −2.7704146441486308228788274919, −1.98481988413404410326300121664, 0.45190471570403847227921128421, 2.024087591724042701739270685688, 2.78984310210347412407134038152, 3.67649014975203508929310705998, 4.95498751956955961782877540134, 5.40488851893341325106530843233, 6.94219758297170639263134037250, 7.71298305455749933342694622009, 8.93695937462579955918998786462, 9.682549222057531438555389314014, 10.549518925903651001967575170301, 12.13326126674803461774947148157, 12.49750463658655776415104456119, 13.40028086241665880432230053994, 13.88935423672073969302026314505, 15.24625481920064341285193464503, 15.68897218418532649489536265939, 16.25249441482655549978349437212, 17.78425109567220333671308160170, 18.91316941224354753669166681933, 19.722940942836554515070933371625, 20.28026333168268691145962685134, 20.89953850291907873925236577970, 21.855066819055180790487891914531, 22.47308604879060321602481663300

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