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Properties

Label 1-569-569.64-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.874 + 0.484i$
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  + (−0.975 − 0.219i)2-s + (−0.759 + 0.650i)3-s + (0.903 + 0.428i)4-s + (−0.598 − 0.801i)5-s + (0.883 − 0.467i)6-s + (0.996 − 0.0883i)7-s + (−0.787 − 0.616i)8-s + (0.154 − 0.988i)9-s + (0.408 + 0.912i)10-s + (−0.814 − 0.580i)11-s + (−0.964 + 0.262i)12-s + (−0.197 − 0.980i)13-s + (−0.991 − 0.132i)14-s + (0.975 + 0.219i)15-s + (0.633 + 0.773i)16-s + (−0.730 − 0.683i)17-s + ⋯
L(s)  = 1  + (−0.975 − 0.219i)2-s + (−0.759 + 0.650i)3-s + (0.903 + 0.428i)4-s + (−0.598 − 0.801i)5-s + (0.883 − 0.467i)6-s + (0.996 − 0.0883i)7-s + (−0.787 − 0.616i)8-s + (0.154 − 0.988i)9-s + (0.408 + 0.912i)10-s + (−0.814 − 0.580i)11-s + (−0.964 + 0.262i)12-s + (−0.197 − 0.980i)13-s + (−0.991 − 0.132i)14-s + (0.975 + 0.219i)15-s + (0.633 + 0.773i)16-s + (−0.730 − 0.683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.874 + 0.484i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.874 + 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008936936958 - 0.03456316736i\)
\(L(\frac12)\) \(\approx\) \(0.008936936958 - 0.03456316736i\)
\(L(1)\) \(\approx\) \(0.4063756423 - 0.06890488370i\)
\(L(1)\) \(\approx\) \(0.4063756423 - 0.06890488370i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.975 - 0.219i)T \)
3 \( 1 + (-0.759 + 0.650i)T \)
5 \( 1 + (-0.598 - 0.801i)T \)
7 \( 1 + (0.996 - 0.0883i)T \)
11 \( 1 + (-0.814 - 0.580i)T \)
13 \( 1 + (-0.197 - 0.980i)T \)
17 \( 1 + (-0.730 - 0.683i)T \)
19 \( 1 + (-0.903 + 0.428i)T \)
23 \( 1 + (-0.699 - 0.714i)T \)
29 \( 1 + (0.730 - 0.683i)T \)
31 \( 1 + (-0.240 + 0.970i)T \)
37 \( 1 + (-0.996 + 0.0883i)T \)
41 \( 1 + (0.699 + 0.714i)T \)
43 \( 1 + (-0.975 + 0.219i)T \)
47 \( 1 + (0.367 + 0.930i)T \)
53 \( 1 + (0.883 - 0.467i)T \)
59 \( 1 + (0.367 + 0.930i)T \)
61 \( 1 + (0.937 - 0.346i)T \)
67 \( 1 + (-0.730 + 0.683i)T \)
71 \( 1 + (-0.787 + 0.616i)T \)
73 \( 1 + (-0.903 + 0.428i)T \)
79 \( 1 + (0.562 + 0.826i)T \)
83 \( 1 + (-0.984 + 0.176i)T \)
89 \( 1 + (-0.240 - 0.970i)T \)
97 \( 1 + (0.283 - 0.958i)T \)
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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

−23.73917553111739040723071178649, −23.42502802645022617404182952858, −22.039411502252014662205603446332, −21.29975320926942941370887980233, −20.075776652446838132435870347694, −19.24884318438391684673210323940, −18.6304653238104079201848246130, −17.79903188003894161812440052081, −17.44296029410052686447092276468, −16.3269911600901705526951250070, −15.41344502877193610140330605917, −14.7514582197383636283980851701, −13.624805323170296747396346243861, −12.22132575033323664304960183660, −11.61197436126523819607491219920, −10.80591427211278357197467073632, −10.30996841349239142209198341610, −8.75818039000617759935058535152, −7.91438045257212472489826140267, −7.18610464004891980243915615728, −6.53718119781407405916361197173, −5.404230090562871383043728142899, −4.26484245066630888962315034731, −2.32551076002812649149910759720, −1.804784978524698879638149078573, 0.030746777689268518164801501115, 1.138684649480215360512143452443, 2.67786201841871187369711189921, 4.0338211912104259591179197476, 4.95388023863635095200531874914, 5.89657558867909150493133941787, 7.21360905103766426938741320956, 8.32083120758355102411606851147, 8.63467693028463394643322811227, 10.040234230013403989470381514235, 10.66717966259383024587178244163, 11.451850267472319598820026916738, 12.15849809566590658408346516029, 13.04350140519286446587251993153, 14.69231973105135120349528690118, 15.665591693199571536214063953532, 16.07751381305020355860243823467, 16.993466170612355947676393304039, 17.725899785741973232132577172696, 18.31877219100413383958346766579, 19.52471916092376539953088966732, 20.42643438534281989480875577823, 20.93075129433435450603418342846, 21.607680037640079920919871301687, 22.81560188158404625897893220744

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