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Properties

Label 1-571-571.38-r0-0-0
Degree $1$
Conductor $571$
Sign $0.897 - 0.440i$
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  + (0.945 + 0.324i)2-s + (0.0495 − 0.998i)3-s + (0.789 + 0.614i)4-s + (0.490 − 0.871i)5-s + (0.371 − 0.928i)6-s + (0.894 + 0.446i)7-s + (0.546 + 0.837i)8-s + (−0.995 − 0.0990i)9-s + (0.746 − 0.665i)10-s + (0.746 + 0.665i)11-s + (0.652 − 0.757i)12-s + (0.180 − 0.983i)13-s + (0.701 + 0.712i)14-s + (−0.846 − 0.533i)15-s + (0.245 + 0.969i)16-s + (0.997 + 0.0660i)17-s + ⋯
L(s)  = 1  + (0.945 + 0.324i)2-s + (0.0495 − 0.998i)3-s + (0.789 + 0.614i)4-s + (0.490 − 0.871i)5-s + (0.371 − 0.928i)6-s + (0.894 + 0.446i)7-s + (0.546 + 0.837i)8-s + (−0.995 − 0.0990i)9-s + (0.746 − 0.665i)10-s + (0.746 + 0.665i)11-s + (0.652 − 0.757i)12-s + (0.180 − 0.983i)13-s + (0.701 + 0.712i)14-s + (−0.846 − 0.533i)15-s + (0.245 + 0.969i)16-s + (0.997 + 0.0660i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.897 - 0.440i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.897 - 0.440i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.002480182 - 0.6961397531i\)
\(L(\frac12)\) \(\approx\) \(3.002480182 - 0.6961397531i\)
\(L(1)\) \(\approx\) \(2.152428169 - 0.3014294166i\)
\(L(1)\) \(\approx\) \(2.152428169 - 0.3014294166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.945 + 0.324i)T \)
3 \( 1 + (0.0495 - 0.998i)T \)
5 \( 1 + (0.490 - 0.871i)T \)
7 \( 1 + (0.894 + 0.446i)T \)
11 \( 1 + (0.746 + 0.665i)T \)
13 \( 1 + (0.180 - 0.983i)T \)
17 \( 1 + (0.997 + 0.0660i)T \)
19 \( 1 + (-0.627 + 0.778i)T \)
23 \( 1 + (-0.934 + 0.355i)T \)
29 \( 1 + (-0.879 - 0.475i)T \)
31 \( 1 + (0.245 + 0.969i)T \)
37 \( 1 + (-0.724 - 0.689i)T \)
41 \( 1 + (-0.677 - 0.735i)T \)
43 \( 1 + (0.863 + 0.504i)T \)
47 \( 1 + (0.245 + 0.969i)T \)
53 \( 1 + (-0.0165 - 0.999i)T \)
59 \( 1 + (-0.986 + 0.164i)T \)
61 \( 1 + (-0.574 - 0.818i)T \)
67 \( 1 + (-0.0165 - 0.999i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.724 - 0.689i)T \)
79 \( 1 + (-0.999 + 0.0330i)T \)
83 \( 1 + (0.371 - 0.928i)T \)
89 \( 1 + (0.371 - 0.928i)T \)
97 \( 1 + (-0.340 + 0.940i)T \)
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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

−23.18621184120330596210689976009, −22.05987325406921827997870651066, −21.85446153293504693920422843963, −21.0103736706104606100648781056, −20.37716631974333807698342514364, −19.3163749682001850380212102721, −18.510547032855104333238364817027, −17.08732694392548775494302377679, −16.60881812127159902448012025477, −15.38443261109745737354875063722, −14.64294445164276421906273035600, −14.10846019513447280077482447194, −13.56147438654014873063092754233, −11.852193045324984785157089019990, −11.33092245105484046564832327999, −10.594632920673952801982660743870, −9.843640976190550351276671645420, −8.7413305565650786252614785537, −7.333107399843391458763955739122, −6.28153734009990149644818341379, −5.51398261712939609666350434355, −4.36314388244481240237352445988, −3.75503922900125389440573166729, −2.687753222707371113096008982615, −1.57165349933753359173642505642, 1.50002445804266574846624063162, 2.015374754797501461228234135725, 3.452424407279369212376476724118, 4.693261459441717214824534838410, 5.66655389556736469046855847664, 6.10737073605682039944792723060, 7.53537260987790514629833378364, 8.058766772308018854498317298131, 8.99848537063721200270391943345, 10.484474192805397499411065873839, 11.7752686613521581993695639875, 12.32848388522465846178747039859, 12.830525847370460178646285189688, 13.98067175674816844075086521919, 14.418756136349839005028887619960, 15.38732443936909375167465721829, 16.554548165277354066467228834219, 17.43926237503374312335110635226, 17.7774978943919050423721980004, 19.122511424076036477077429590913, 20.18572012438955114085704523599, 20.699743911773376494883265769326, 21.49905448878997782788628749035, 22.60203123145910794825884822728, 23.25810958206867029354641094541

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