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Properties

Label 1-569-569.162-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.399 - 0.916i$
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  + (−0.996 − 0.0883i)2-s + (0.826 − 0.562i)3-s + (0.984 + 0.176i)4-s + (0.633 − 0.773i)5-s + (−0.873 + 0.487i)6-s + (0.787 − 0.616i)7-s + (−0.964 − 0.262i)8-s + (0.367 − 0.930i)9-s + (−0.699 + 0.714i)10-s + (−0.997 + 0.0663i)11-s + (0.912 − 0.408i)12-s + (−0.759 + 0.650i)13-s + (−0.839 + 0.544i)14-s + (0.0883 − 0.996i)15-s + (0.937 + 0.346i)16-s + (0.598 + 0.801i)17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0883i)2-s + (0.826 − 0.562i)3-s + (0.984 + 0.176i)4-s + (0.633 − 0.773i)5-s + (−0.873 + 0.487i)6-s + (0.787 − 0.616i)7-s + (−0.964 − 0.262i)8-s + (0.367 − 0.930i)9-s + (−0.699 + 0.714i)10-s + (−0.997 + 0.0663i)11-s + (0.912 − 0.408i)12-s + (−0.759 + 0.650i)13-s + (−0.839 + 0.544i)14-s + (0.0883 − 0.996i)15-s + (0.937 + 0.346i)16-s + (0.598 + 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.399 - 0.916i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (162, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.399 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6976461851 - 1.065512172i\)
\(L(\frac12)\) \(\approx\) \(0.6976461851 - 1.065512172i\)
\(L(1)\) \(\approx\) \(0.8849670584 - 0.4982533911i\)
\(L(1)\) \(\approx\) \(0.8849670584 - 0.4982533911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.996 - 0.0883i)T \)
3 \( 1 + (0.826 - 0.562i)T \)
5 \( 1 + (0.633 - 0.773i)T \)
7 \( 1 + (0.787 - 0.616i)T \)
11 \( 1 + (-0.997 + 0.0663i)T \)
13 \( 1 + (-0.759 + 0.650i)T \)
17 \( 1 + (0.598 + 0.801i)T \)
19 \( 1 + (-0.176 - 0.984i)T \)
23 \( 1 + (-0.304 - 0.952i)T \)
29 \( 1 + (-0.801 - 0.598i)T \)
31 \( 1 + (0.506 + 0.862i)T \)
37 \( 1 + (-0.616 - 0.787i)T \)
41 \( 1 + (-0.952 + 0.304i)T \)
43 \( 1 + (0.996 - 0.0883i)T \)
47 \( 1 + (0.894 - 0.448i)T \)
53 \( 1 + (0.873 - 0.487i)T \)
59 \( 1 + (-0.894 + 0.448i)T \)
61 \( 1 + (0.883 + 0.467i)T \)
67 \( 1 + (0.598 - 0.801i)T \)
71 \( 1 + (-0.964 + 0.262i)T \)
73 \( 1 + (0.176 + 0.984i)T \)
79 \( 1 + (0.525 + 0.850i)T \)
83 \( 1 + (0.970 + 0.240i)T \)
89 \( 1 + (-0.506 + 0.862i)T \)
97 \( 1 + (0.980 - 0.197i)T \)
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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

−23.81552408152496947595786497048, −22.405457209296382193284136652502, −21.590422701802454001319800272174, −20.787878057308925866274773100396, −20.41342186612497151261898421641, −18.9760791403535226186737546072, −18.69285390969800657781093836241, −17.79099780247758584970413854857, −16.94771794612557257756968701637, −15.80323738099585266756572937921, −15.14332041437463230338267945659, −14.536417023929396942242301296723, −13.621854364928034188725106965825, −12.24773453327013156579975187634, −11.144612412344081189421530516839, −10.28900803750573128288602165660, −9.798283703623722094731757739616, −8.85253104610725828706834165480, −7.75449325204766194536032310401, −7.48396720310325583320952282998, −5.79635470681722538419385492182, −5.15905226634098841389008865018, −3.29895526267871929164683506672, −2.524475486564909528066600741600, −1.76333052378475634920709821292, 0.79027669639095777109424615263, 1.90989613841521185332817204057, 2.4987304973242378113144136348, 4.07165132888482809639066334164, 5.33895923317436579438305606057, 6.659963385610141788276736906667, 7.51674205845616058546876646393, 8.28654235601372802389210200566, 8.94062873024867568878142900112, 9.93184383016906318797640796888, 10.6454200895845011094752364140, 11.97773096339513508658681308522, 12.69705334838267595719807620977, 13.64269475392978036671193954324, 14.534493388262249282789623280370, 15.43823306304123393123247815701, 16.57676661622765135396231913029, 17.328492671409104826720846343415, 17.934823113700942670025456469292, 18.8396139245701203495311256451, 19.67272236750621930280612087046, 20.40256457552937468101221083678, 21.0596237722434746373224578053, 21.564228514070376237302234751536, 23.53993502612593375695945268963

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