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Properties

Label 1-569-569.111-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.999 + 0.0329i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (−0.991 + 0.132i)2-s + (−0.110 + 0.993i)3-s + (0.964 − 0.262i)4-s + (0.240 + 0.970i)5-s + (−0.0221 − 0.999i)6-s + (−0.839 + 0.544i)7-s + (−0.921 + 0.387i)8-s + (−0.975 − 0.219i)9-s + (−0.367 − 0.930i)10-s + (0.633 + 0.773i)11-s + (0.154 + 0.988i)12-s + (0.487 + 0.873i)13-s + (0.759 − 0.650i)14-s + (−0.991 + 0.132i)15-s + (0.862 − 0.506i)16-s + (0.984 + 0.176i)17-s + ⋯
L(s)  = 1  + (−0.991 + 0.132i)2-s + (−0.110 + 0.993i)3-s + (0.964 − 0.262i)4-s + (0.240 + 0.970i)5-s + (−0.0221 − 0.999i)6-s + (−0.839 + 0.544i)7-s + (−0.921 + 0.387i)8-s + (−0.975 − 0.219i)9-s + (−0.367 − 0.930i)10-s + (0.633 + 0.773i)11-s + (0.154 + 0.988i)12-s + (0.487 + 0.873i)13-s + (0.759 − 0.650i)14-s + (−0.991 + 0.132i)15-s + (0.862 − 0.506i)16-s + (0.984 + 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.999 + 0.0329i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.999 + 0.0329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01316337233 + 0.7995618603i\)
\(L(\frac12)\) \(\approx\) \(0.01316337233 + 0.7995618603i\)
\(L(1)\) \(\approx\) \(0.4872550023 + 0.4995688328i\)
\(L(1)\) \(\approx\) \(0.4872550023 + 0.4995688328i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.991 + 0.132i)T \)
3 \( 1 + (-0.110 + 0.993i)T \)
5 \( 1 + (0.240 + 0.970i)T \)
7 \( 1 + (-0.839 + 0.544i)T \)
11 \( 1 + (0.633 + 0.773i)T \)
13 \( 1 + (0.487 + 0.873i)T \)
17 \( 1 + (0.984 + 0.176i)T \)
19 \( 1 + (0.964 + 0.262i)T \)
23 \( 1 + (-0.448 + 0.894i)T \)
29 \( 1 + (0.984 - 0.176i)T \)
31 \( 1 + (0.699 + 0.714i)T \)
37 \( 1 + (-0.839 + 0.544i)T \)
41 \( 1 + (-0.448 + 0.894i)T \)
43 \( 1 + (-0.991 - 0.132i)T \)
47 \( 1 + (0.996 - 0.0883i)T \)
53 \( 1 + (-0.0221 - 0.999i)T \)
59 \( 1 + (0.996 - 0.0883i)T \)
61 \( 1 + (-0.666 - 0.745i)T \)
67 \( 1 + (0.984 - 0.176i)T \)
71 \( 1 + (-0.921 - 0.387i)T \)
73 \( 1 + (0.964 + 0.262i)T \)
79 \( 1 + (-0.999 - 0.0442i)T \)
83 \( 1 + (0.408 - 0.912i)T \)
89 \( 1 + (0.699 - 0.714i)T \)
97 \( 1 + (-0.883 + 0.467i)T \)
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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

−23.00699448491498288313050444149, −22.01038670710826336471402072823, −20.69425025209163142107932075684, −20.18401845983050885291118195158, −19.44020040324195073956024284980, −18.727734913274385450867179551729, −17.81833420763141774575611350355, −17.02941451689032540076342500258, −16.46443444980762431012366417100, −15.71476788356097352502199423958, −14.07118566490988435511833784717, −13.386514997171146270924151784689, −12.35462300435037039041299792570, −11.925501483970395818193015138658, −10.68206677044769405514972043370, −9.80353970246845962980241210368, −8.78207784021472955701215376366, −8.15989347876938946090693559655, −7.19985996651197473979912693469, −6.23626219021327986182952343371, −5.537184642395168587458451366661, −3.625777010731456374340102263349, −2.65489315198380504550519213098, −1.1426400272858700595891544051, −0.69780597367433666109889868563, 1.65929488730485397821066902531, 2.96805465275315488127513552843, 3.63620494084825840936121213745, 5.31956400715020079945397751349, 6.29565298758599493937610023903, 6.89103631767669206109106158820, 8.20513550624339820853431125567, 9.28950642024320818745778018597, 9.87085668096759542864560323341, 10.342185562920377155667030717887, 11.70577064327325375708192490154, 11.92599567567697807455780065124, 13.88529651648628574381486190190, 14.63963641591846742731991299668, 15.52705191878169835220481859364, 16.048690381925966834981765540884, 16.98158085905146210009609008137, 17.7757766243449622118008024392, 18.663133718385094626898961781231, 19.392844783750205172472170023241, 20.2005240963342887613363902308, 21.283376043149709769895594755112, 21.81795462631433379658311063603, 22.8136533625728163151600751585, 23.45474107921842746191955254499

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