Properties

Label 1-569-569.112-r0-0-0
Degree $1$
Conductor $569$
Sign $0.995 - 0.0905i$
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 + (0.993 − 0.110i)3-s + (0.964 + 0.262i)4-s + (0.240 − 0.970i)5-s + (0.999 + 0.0221i)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 + (−0.773 − 0.633i)11-s + (0.988 + 0.154i)12-s + (−0.487 + 0.873i)13-s + (0.759 + 0.650i)14-s + (0.132 − 0.991i)15-s + (0.862 + 0.506i)16-s + (−0.984 + 0.176i)17-s + ⋯
L(s)  = 1  + (0.991 + 0.132i)2-s + (0.993 − 0.110i)3-s + (0.964 + 0.262i)4-s + (0.240 − 0.970i)5-s + (0.999 + 0.0221i)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 + (−0.773 − 0.633i)11-s + (0.988 + 0.154i)12-s + (−0.487 + 0.873i)13-s + (0.759 + 0.650i)14-s + (0.132 − 0.991i)15-s + (0.862 + 0.506i)16-s + (−0.984 + 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.995 - 0.0905i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.995 - 0.0905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.921364432 - 0.1778664046i\)
\(L(\frac12)\) \(\approx\) \(3.921364432 - 0.1778664046i\)
\(L(1)\) \(\approx\) \(2.697362720 - 0.06125783972i\)
\(L(1)\) \(\approx\) \(2.697362720 - 0.06125783972i\)

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

−23.31793952707675565291122556607, −22.14927236942106375086069929537, −21.796593518948434159254759383572, −20.69362542776521128111501556392, −20.16613927655053683394998823278, −19.52624126084562153136713290112, −18.19112571389750970770342466292, −17.69169226509278322490955984885, −16.06572486227115167944136309991, −15.30100521307858362952074260036, −14.69840404909759193299222198111, −14.03217645239793770868000076223, −13.29929662863667870349529135485, −12.46963608506459886270727907572, −11.077329812907547205261637520377, −10.556730149595349694031560577817, −9.73150824803255461251355317245, −8.198432383718649501012066307464, −7.330515228792158104706357462488, −6.79933891426792030478367361001, −5.222588385354207277899283985486, −4.51179727829978658682331138047, −3.36050881910294781477186956508, −2.546988499814878256589443121, −1.783928448439653456099817563841, 1.79792290216094607246889970175, 2.174318096691520583107753494834, 3.61209021441858863209409118843, 4.5244020211992326184080864625, 5.322420950471456246914599462, 6.35726614926378174072939299150, 7.745952418210398184965181735151, 8.230057482495501792045749776939, 9.19332000389296444358082633468, 10.34773825973494413014716820993, 11.69338416224418617806837826345, 12.24556735150054510133083174896, 13.41861616648369984194630945167, 13.70165586360578148691645116524, 14.70027107875704561066195741694, 15.493656938736601744625752860882, 16.210422053044743598097619992066, 17.189385739144535594131880262007, 18.36860607866844922378634100135, 19.32078948128275321535380879378, 20.29419576104299151324949804133, 20.8285236188550587636651564644, 21.48989870181652907493195733309, 22.08654989040396786175654515911, 23.74038103148871294418427597370

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