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Properties

Label 1-571-571.189-r0-0-0
Degree $1$
Conductor $571$
Sign $0.656 - 0.754i$
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  + (−0.0825 − 0.996i)2-s + (−0.846 + 0.533i)3-s + (−0.986 + 0.164i)4-s + (0.0495 + 0.998i)5-s + (0.601 + 0.799i)6-s + (0.490 − 0.871i)7-s + (0.245 + 0.969i)8-s + (0.431 − 0.901i)9-s + (0.991 − 0.131i)10-s + (0.991 + 0.131i)11-s + (0.746 − 0.665i)12-s + (−0.999 − 0.0330i)13-s + (−0.909 − 0.416i)14-s + (−0.574 − 0.818i)15-s + (0.945 − 0.324i)16-s + (−0.956 − 0.293i)17-s + ⋯
L(s)  = 1  + (−0.0825 − 0.996i)2-s + (−0.846 + 0.533i)3-s + (−0.986 + 0.164i)4-s + (0.0495 + 0.998i)5-s + (0.601 + 0.799i)6-s + (0.490 − 0.871i)7-s + (0.245 + 0.969i)8-s + (0.431 − 0.901i)9-s + (0.991 − 0.131i)10-s + (0.991 + 0.131i)11-s + (0.746 − 0.665i)12-s + (−0.999 − 0.0330i)13-s + (−0.909 − 0.416i)14-s + (−0.574 − 0.818i)15-s + (0.945 − 0.324i)16-s + (−0.956 − 0.293i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.656 - 0.754i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.656 - 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7981424119 - 0.3635415568i\)
\(L(\frac12)\) \(\approx\) \(0.7981424119 - 0.3635415568i\)
\(L(1)\) \(\approx\) \(0.7406570574 - 0.2126189309i\)
\(L(1)\) \(\approx\) \(0.7406570574 - 0.2126189309i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.0825 - 0.996i)T \)
3 \( 1 + (-0.846 + 0.533i)T \)
5 \( 1 + (0.0495 + 0.998i)T \)
7 \( 1 + (0.490 - 0.871i)T \)
11 \( 1 + (0.991 + 0.131i)T \)
13 \( 1 + (-0.999 - 0.0330i)T \)
17 \( 1 + (-0.956 - 0.293i)T \)
19 \( 1 + (-0.768 - 0.639i)T \)
23 \( 1 + (0.997 - 0.0660i)T \)
29 \( 1 + (0.789 + 0.614i)T \)
31 \( 1 + (0.945 - 0.324i)T \)
37 \( 1 + (-0.277 + 0.960i)T \)
41 \( 1 + (0.546 - 0.837i)T \)
43 \( 1 + (-0.724 + 0.689i)T \)
47 \( 1 + (0.945 - 0.324i)T \)
53 \( 1 + (0.652 - 0.757i)T \)
59 \( 1 + (-0.677 - 0.735i)T \)
61 \( 1 + (0.922 - 0.386i)T \)
67 \( 1 + (0.652 - 0.757i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.277 + 0.960i)T \)
79 \( 1 + (-0.148 - 0.988i)T \)
83 \( 1 + (0.601 + 0.799i)T \)
89 \( 1 + (0.601 + 0.799i)T \)
97 \( 1 + (0.701 - 0.712i)T \)
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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

−23.55176934663026584182223856284, −22.74016588310717395487729145100, −21.844422067981685334661630536845, −21.28422214280605186188736468253, −19.60948997164883747525479233187, −19.11682793504267911777169179629, −17.997385934311673412084182840783, −17.200881725081597179098469808141, −16.98667126424897481403942908312, −15.90111753197455512864868686951, −15.09858995265057527622337782785, −14.10777095306639293237793891308, −13.09386826751306896895721869708, −12.34244253248609197538803107634, −11.72111685743728507224018508905, −10.354299372743882421329928589318, −9.14538663015968416200027035766, −8.55619118871092058274689757737, −7.58837449987017335166724161803, −6.50728494435218691835572918097, −5.82398937290234394195844349255, −4.87213849982882374046350198630, −4.31748365354564691995704007854, −2.08273169944953384977692840339, −0.90803429837138678547305606418, 0.75091667712454827016926958102, 2.17758053821218790300542647520, 3.377642950982001451538669792149, 4.383734071787682750586098815699, 4.93811391991630405529241097538, 6.531215456491436998152554334164, 7.16028228416328788287777689645, 8.677824567297616915066273321564, 9.74460471311120784510592934474, 10.369296800945115149738863966017, 11.183992448086066423994481626924, 11.60190793837285254461335078794, 12.68139361081426436548552590723, 13.80188372766653754603662543169, 14.59833558326036005849123925273, 15.36499341496165397791402592848, 16.971185626165703601043392414818, 17.32686423398624844490055799073, 17.97888002866482248986977539834, 19.10408553391628280035834303775, 19.81389498565178992168637087677, 20.71689904294461844802910115051, 21.71243511935553265809666489511, 22.11224233791289660117432574407, 22.90835052521920802245483125886

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