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Properties

Label 1-569-569.158-r0-0-0
Degree $1$
Conductor $569$
Sign $0.765 + 0.643i$
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  + (−0.197 + 0.980i)2-s + (−0.814 − 0.580i)3-s + (−0.921 − 0.387i)4-s + (0.408 + 0.912i)5-s + (0.730 − 0.683i)6-s + (0.759 − 0.650i)7-s + (0.562 − 0.826i)8-s + (0.325 + 0.945i)9-s + (−0.975 + 0.219i)10-s + (−0.240 + 0.970i)11-s + (0.525 + 0.850i)12-s + (−0.0221 − 0.999i)13-s + (0.487 + 0.873i)14-s + (0.197 − 0.980i)15-s + (0.699 + 0.714i)16-s + (0.964 − 0.262i)17-s + ⋯
L(s)  = 1  + (−0.197 + 0.980i)2-s + (−0.814 − 0.580i)3-s + (−0.921 − 0.387i)4-s + (0.408 + 0.912i)5-s + (0.730 − 0.683i)6-s + (0.759 − 0.650i)7-s + (0.562 − 0.826i)8-s + (0.325 + 0.945i)9-s + (−0.975 + 0.219i)10-s + (−0.240 + 0.970i)11-s + (0.525 + 0.850i)12-s + (−0.0221 − 0.999i)13-s + (0.487 + 0.873i)14-s + (0.197 − 0.980i)15-s + (0.699 + 0.714i)16-s + (0.964 − 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.765 + 0.643i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.765 + 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9413866542 + 0.3432144207i\)
\(L(\frac12)\) \(\approx\) \(0.9413866542 + 0.3432144207i\)
\(L(1)\) \(\approx\) \(0.7938468383 + 0.2622067748i\)
\(L(1)\) \(\approx\) \(0.7938468383 + 0.2622067748i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.197 + 0.980i)T \)
3 \( 1 + (-0.814 - 0.580i)T \)
5 \( 1 + (0.408 + 0.912i)T \)
7 \( 1 + (0.759 - 0.650i)T \)
11 \( 1 + (-0.240 + 0.970i)T \)
13 \( 1 + (-0.0221 - 0.999i)T \)
17 \( 1 + (0.964 - 0.262i)T \)
19 \( 1 + (0.921 - 0.387i)T \)
23 \( 1 + (-0.996 - 0.0883i)T \)
29 \( 1 + (-0.964 - 0.262i)T \)
31 \( 1 + (0.367 - 0.930i)T \)
37 \( 1 + (-0.759 + 0.650i)T \)
41 \( 1 + (0.996 + 0.0883i)T \)
43 \( 1 + (-0.197 - 0.980i)T \)
47 \( 1 + (0.991 + 0.132i)T \)
53 \( 1 + (0.730 - 0.683i)T \)
59 \( 1 + (0.991 + 0.132i)T \)
61 \( 1 + (-0.952 - 0.304i)T \)
67 \( 1 + (0.964 + 0.262i)T \)
71 \( 1 + (0.562 + 0.826i)T \)
73 \( 1 + (0.921 - 0.387i)T \)
79 \( 1 + (0.0663 + 0.997i)T \)
83 \( 1 + (-0.154 + 0.988i)T \)
89 \( 1 + (0.367 + 0.930i)T \)
97 \( 1 + (0.666 - 0.745i)T \)
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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

−22.999407224268149391546349859671, −21.90727148983293437363012393631, −21.34877901517526227621573465078, −21.04459351349101735470892237605, −20.06592617848311882209731910975, −18.85772231349280953985223889174, −18.18702851715109166825511128292, −17.39249594317898965197377000367, −16.5392034591688909206616175837, −15.99944410303496698917049960856, −14.4432477643416993611348871016, −13.77973933547081278118784226506, −12.46556787423071267494211452034, −11.99443949117858586170534304300, −11.25466359291728275778583253165, −10.31967144857346072157442206010, −9.38972657206295354086909012690, −8.79241041629069679910937212863, −7.77505499884031828314797653009, −5.83796850445162666790681812042, −5.36712611972659390440123085512, −4.41979470065047578393469193203, −3.4494399474710906409970676301, −1.91319649789802585065218467360, −0.98630190203169401259074631001, 0.86685631389804764116217369144, 2.16394043726688733504799735165, 3.8941774513127148441800055798, 5.18282845241738268812618120671, 5.672910935229972305529926945532, 6.84150517172534130520967297319, 7.51372230550162340451403187738, 7.97052665937512468354923007481, 9.80521499934471093743592546042, 10.249107772344375026845841418734, 11.23444098658293416350250646493, 12.35685207701582555523395617993, 13.45042090047638848134486353726, 14.02731656046861634379226186918, 14.96951165373959427701799137574, 15.76455354598532945261775188983, 16.99026420281726226228121132925, 17.44764344413736209564887029148, 18.20486865157382188751871604727, 18.578182187823797451267293369843, 19.84440627405373018057294699751, 20.94476823911754076676898663003, 22.36931306740949921100436093341, 22.54679803026876619806835841988, 23.403921237004722629075667809318

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