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Properties

Label 1-569-569.100-r0-0-0
Degree $1$
Conductor $569$
Sign $0.991 + 0.133i$
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  + (−0.787 + 0.616i)2-s + (0.525 − 0.850i)3-s + (0.240 − 0.970i)4-s + (0.937 + 0.346i)5-s + (0.110 + 0.993i)6-s + (0.964 + 0.262i)7-s + (0.408 + 0.912i)8-s + (−0.448 − 0.894i)9-s + (−0.952 + 0.304i)10-s + (0.283 + 0.958i)11-s + (−0.699 − 0.714i)12-s + (0.562 − 0.826i)13-s + (−0.921 + 0.387i)14-s + (0.787 − 0.616i)15-s + (−0.883 − 0.467i)16-s + (0.633 + 0.773i)17-s + ⋯
L(s)  = 1  + (−0.787 + 0.616i)2-s + (0.525 − 0.850i)3-s + (0.240 − 0.970i)4-s + (0.937 + 0.346i)5-s + (0.110 + 0.993i)6-s + (0.964 + 0.262i)7-s + (0.408 + 0.912i)8-s + (−0.448 − 0.894i)9-s + (−0.952 + 0.304i)10-s + (0.283 + 0.958i)11-s + (−0.699 − 0.714i)12-s + (0.562 − 0.826i)13-s + (−0.921 + 0.387i)14-s + (0.787 − 0.616i)15-s + (−0.883 − 0.467i)16-s + (0.633 + 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.991 + 0.133i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.991 + 0.133i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.567830363 + 0.1048682445i\)
\(L(\frac12)\) \(\approx\) \(1.567830363 + 0.1048682445i\)
\(L(1)\) \(\approx\) \(1.170414374 + 0.06523149867i\)
\(L(1)\) \(\approx\) \(1.170414374 + 0.06523149867i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.787 + 0.616i)T \)
3 \( 1 + (0.525 - 0.850i)T \)
5 \( 1 + (0.937 + 0.346i)T \)
7 \( 1 + (0.964 + 0.262i)T \)
11 \( 1 + (0.283 + 0.958i)T \)
13 \( 1 + (0.562 - 0.826i)T \)
17 \( 1 + (0.633 + 0.773i)T \)
19 \( 1 + (-0.240 - 0.970i)T \)
23 \( 1 + (0.730 + 0.683i)T \)
29 \( 1 + (-0.633 + 0.773i)T \)
31 \( 1 + (0.666 + 0.745i)T \)
37 \( 1 + (-0.964 - 0.262i)T \)
41 \( 1 + (-0.730 - 0.683i)T \)
43 \( 1 + (-0.787 - 0.616i)T \)
47 \( 1 + (-0.903 + 0.428i)T \)
53 \( 1 + (0.110 + 0.993i)T \)
59 \( 1 + (-0.903 + 0.428i)T \)
61 \( 1 + (0.487 + 0.873i)T \)
67 \( 1 + (0.633 - 0.773i)T \)
71 \( 1 + (0.408 - 0.912i)T \)
73 \( 1 + (-0.240 - 0.970i)T \)
79 \( 1 + (-0.975 - 0.219i)T \)
83 \( 1 + (-0.862 - 0.506i)T \)
89 \( 1 + (0.666 - 0.745i)T \)
97 \( 1 + (-0.759 - 0.650i)T \)
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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

−22.99904809930485878314727829933, −21.9212828151964497608958833447, −21.10933405463169029582140653204, −20.9500936240930894631034207655, −20.20775650584352037486593344089, −18.91809042694360465328245902637, −18.49757345594329024093835274285, −17.10152201334403402025633523083, −16.81104396864240291102470399625, −16.01350793097832821230250535287, −14.617520126976163820718563270253, −13.91183732849641421253868307738, −13.18167785584337282786622201192, −11.68519921545106775899500644392, −11.13597257459809431584060102070, −10.11289718182437884088210246689, −9.53210388096386385174290022016, −8.49276232860725587152175991605, −8.18316305235372017563512917030, −6.64204443079086481287141723, −5.32480126108038707148690539348, −4.291734498888006374973714771362, −3.30907582395651302551479926082, −2.15477560394208638049818758923, −1.234621665881230136956584495569, 1.382168758767326666892234686275, 1.827015635614745863886239370986, 3.08538636289299660374923160056, 5.019068291345261749265507938819, 5.83477523384172269488669367971, 6.84686493306139961724179027011, 7.489155418791829412629285869217, 8.53998655686008823443183545807, 9.10334077575975147755882350696, 10.2221158105485887394993487065, 11.02952148712702070733385428707, 12.23541099405656061895836243380, 13.321895642045640476572765126742, 14.116444840993244900878517333013, 14.95430698763089244665193959580, 15.33673019992186713006246620120, 17.12441113786316333840817863271, 17.511764482889414649766651361522, 18.11751449308427556903585261653, 18.83537327317396461924836694413, 19.80182277455529660647220939489, 20.58569358912054046098163724274, 21.387428154625284018207331888500, 22.75555788376246220828821518059, 23.56191713925577453103180675792

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