Properties

Label 1-569-569.62-r1-0-0
Degree $1$
Conductor $569$
Sign $-0.880 + 0.474i$
Analytic cond. $61.1475$
Root an. cond. $61.1475$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.650 + 0.759i)2-s + (−0.658 − 0.752i)3-s + (−0.154 + 0.988i)4-s + (0.699 + 0.714i)5-s + (0.143 − 0.989i)6-s + (0.826 + 0.562i)7-s + (−0.850 + 0.525i)8-s + (−0.132 + 0.991i)9-s + (−0.0883 + 0.996i)10-s + (0.251 + 0.967i)11-s + (0.845 − 0.534i)12-s + (0.580 + 0.814i)13-s + (0.110 + 0.993i)14-s + (0.0773 − 0.997i)15-s + (−0.952 − 0.304i)16-s + (0.912 + 0.408i)17-s + ⋯
L(s)  = 1  + (0.650 + 0.759i)2-s + (−0.658 − 0.752i)3-s + (−0.154 + 0.988i)4-s + (0.699 + 0.714i)5-s + (0.143 − 0.989i)6-s + (0.826 + 0.562i)7-s + (−0.850 + 0.525i)8-s + (−0.132 + 0.991i)9-s + (−0.0883 + 0.996i)10-s + (0.251 + 0.967i)11-s + (0.845 − 0.534i)12-s + (0.580 + 0.814i)13-s + (0.110 + 0.993i)14-s + (0.0773 − 0.997i)15-s + (−0.952 − 0.304i)16-s + (0.912 + 0.408i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.880 + 0.474i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ -0.880 + 0.474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7434946835 + 2.945337884i\)
\(L(\frac12)\) \(\approx\) \(0.7434946835 + 2.945337884i\)
\(L(1)\) \(\approx\) \(1.209656817 + 0.9974526684i\)
\(L(1)\) \(\approx\) \(1.209656817 + 0.9974526684i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.650 + 0.759i)T \)
3 \( 1 + (-0.658 - 0.752i)T \)
5 \( 1 + (0.699 + 0.714i)T \)
7 \( 1 + (0.826 + 0.562i)T \)
11 \( 1 + (0.251 + 0.967i)T \)
13 \( 1 + (0.580 + 0.814i)T \)
17 \( 1 + (0.912 + 0.408i)T \)
19 \( 1 + (0.807 + 0.589i)T \)
23 \( 1 + (-0.992 - 0.121i)T \)
29 \( 1 + (0.356 - 0.934i)T \)
31 \( 1 + (-0.315 + 0.948i)T \)
37 \( 1 + (0.982 - 0.186i)T \)
41 \( 1 + (0.787 - 0.616i)T \)
43 \( 1 + (-0.759 - 0.650i)T \)
47 \( 1 + (-0.208 - 0.977i)T \)
53 \( 1 + (0.989 + 0.143i)T \)
59 \( 1 + (-0.977 + 0.208i)T \)
61 \( 1 + (-0.683 - 0.730i)T \)
67 \( 1 + (0.912 - 0.408i)T \)
71 \( 1 + (0.850 + 0.525i)T \)
73 \( 1 + (-0.589 + 0.807i)T \)
79 \( 1 + (0.958 - 0.283i)T \)
83 \( 1 + (0.397 - 0.917i)T \)
89 \( 1 + (0.948 - 0.315i)T \)
97 \( 1 + (-0.722 + 0.691i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.49876213890997797702877045741, −21.7015334858136333399481190404, −21.178092000985161793057348123554, −20.40919236117556456393330876366, −19.91016878449804116324774183770, −18.22877061969776861943380431706, −17.89461493278284023231813254576, −16.65573434429699123744474686354, −16.10126381578785861377086431944, −14.9088194064879758404850525027, −14.04537166346414956238084566353, −13.39187990846977612476769825641, −12.3189454195745304481273322358, −11.44278668686405646603076330171, −10.830356802808730862172043943, −9.92244232814830142446878279090, −9.21724977055683090543686586877, −8.02696178316663800188148919849, −6.25634120488510052689804279888, −5.56033471402273575878482335785, −4.898859078345963547849951222954, −3.9489054136581562637644780377, −2.95199227992960559900647674295, −1.23949153398490200180410451589, −0.722593949473756286216855813800, 1.58155798652213808713960761656, 2.37653345731635422115759838770, 3.88741234824669341209528520994, 5.10725711424426357262465514674, 5.83275555185553436057025413547, 6.54700095144983319815716757966, 7.45058445770771841649336401305, 8.20327503960919139281047407934, 9.50483211106087681741110024852, 10.71794299485222308218136739153, 11.90295344707284204915648716177, 12.14660804804277118750755522362, 13.41190442527812655197866100425, 14.16355806416602287300628198606, 14.67767962852701886223791176322, 15.8178842693394917359155988293, 16.80946144654670444344991391834, 17.54577158040745990111584312282, 18.23457263929925618493233652674, 18.6913374580543548850339439126, 20.25929853264102281674080987651, 21.48387725346205306723767920210, 21.71570083047431661639615561982, 22.94034038786271502835660636239, 23.19253563985246766953767115124

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