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Properties

Label 1-569-569.79-r0-0-0
Degree $1$
Conductor $569$
Sign $0.999 + 0.0293i$
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  + (−0.633 + 0.773i)2-s + (−0.304 − 0.952i)3-s + (−0.197 − 0.980i)4-s + (−0.839 + 0.544i)5-s + (0.930 + 0.367i)6-s + (−0.937 − 0.346i)7-s + (0.883 + 0.467i)8-s + (−0.814 + 0.580i)9-s + (0.110 − 0.993i)10-s + (−0.616 + 0.787i)11-s + (−0.873 + 0.487i)12-s + (−0.699 − 0.714i)13-s + (0.862 − 0.506i)14-s + (0.773 + 0.633i)15-s + (−0.921 + 0.387i)16-s + (0.991 + 0.132i)17-s + ⋯
L(s)  = 1  + (−0.633 + 0.773i)2-s + (−0.304 − 0.952i)3-s + (−0.197 − 0.980i)4-s + (−0.839 + 0.544i)5-s + (0.930 + 0.367i)6-s + (−0.937 − 0.346i)7-s + (0.883 + 0.467i)8-s + (−0.814 + 0.580i)9-s + (0.110 − 0.993i)10-s + (−0.616 + 0.787i)11-s + (−0.873 + 0.487i)12-s + (−0.699 − 0.714i)13-s + (0.862 − 0.506i)14-s + (0.773 + 0.633i)15-s + (−0.921 + 0.387i)16-s + (0.991 + 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.999 + 0.0293i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ 0.999 + 0.0293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3881777063 + 0.005699810010i\)
\(L(\frac12)\) \(\approx\) \(0.3881777063 + 0.005699810010i\)
\(L(1)\) \(\approx\) \(0.4593450769 + 0.03740217870i\)
\(L(1)\) \(\approx\) \(0.4593450769 + 0.03740217870i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.633 + 0.773i)T \)
3 \( 1 + (-0.304 - 0.952i)T \)
5 \( 1 + (-0.839 + 0.544i)T \)
7 \( 1 + (-0.937 - 0.346i)T \)
11 \( 1 + (-0.616 + 0.787i)T \)
13 \( 1 + (-0.699 - 0.714i)T \)
17 \( 1 + (0.991 + 0.132i)T \)
19 \( 1 + (-0.980 - 0.197i)T \)
23 \( 1 + (0.0442 + 0.999i)T \)
29 \( 1 + (0.132 + 0.991i)T \)
31 \( 1 + (-0.826 - 0.562i)T \)
37 \( 1 + (0.346 - 0.937i)T \)
41 \( 1 + (-0.999 + 0.0442i)T \)
43 \( 1 + (0.633 + 0.773i)T \)
47 \( 1 + (0.997 - 0.0663i)T \)
53 \( 1 + (-0.930 - 0.367i)T \)
59 \( 1 + (-0.997 + 0.0663i)T \)
61 \( 1 + (-0.154 - 0.988i)T \)
67 \( 1 + (0.991 - 0.132i)T \)
71 \( 1 + (0.883 - 0.467i)T \)
73 \( 1 + (0.980 + 0.197i)T \)
79 \( 1 + (0.730 - 0.683i)T \)
83 \( 1 + (0.650 - 0.759i)T \)
89 \( 1 + (0.826 - 0.562i)T \)
97 \( 1 + (-0.912 - 0.408i)T \)
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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

−23.057595130821539682288687107269, −22.17743693207361066351025128773, −21.47912851719572631940169132479, −20.75965073598477921674011508947, −19.968744248162894202866188022393, −19.01898344973222874105628345766, −18.68193856584129482216870937691, −17.008533857125488083544991897679, −16.67700457244562664813508157200, −15.98635727130020190915820751156, −15.12473975016815809098205618523, −13.79900611986908695963346848803, −12.47262661898510832643455795387, −12.16904575652033946280939339161, −11.11849488207779346076998801643, −10.34393621146623074765461173213, −9.47978610470715623481703327666, −8.738001309693195477630403649236, −7.94773228388425085819337395747, −6.59456336184353336112154432955, −5.26966150587091079104505943701, −4.24418981022366335306925771637, −3.45264022426868834068795856856, −2.52745723429905410639937675165, −0.565081333542382196225406676419, 0.516284562746785279789002059334, 2.10262917661094999317603198417, 3.35188773207038300741756869467, 4.89161839022658931487163875224, 5.91626537326169955098880467988, 6.855056389127770187509771366341, 7.52851037972685128772397702667, 7.95774632353325377688157523357, 9.35422589175824416437568199467, 10.37329894277556144735443505101, 11.01909264481205076111921711020, 12.39127156744609476652435132368, 12.91422790645546251033181183454, 14.13117227712818518438454566163, 14.96433835610399837852693870991, 15.73652199628994257069497514280, 16.692244510949523697087572705623, 17.39899645415888028666567045653, 18.30644716784766614618750578272, 18.94273630661859648774936623451, 19.678456211686413680839149021690, 20.15434552393895508249266021895, 22.0429755111978572827951094468, 22.921596040681177829851128427617, 23.408306092927671078917444542111

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