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Properties

Label 1-569-569.53-r1-0-0
Degree $1$
Conductor $569$
Sign $0.973 - 0.227i$
Analytic cond. $61.1475$
Root an. cond. $61.1475$
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  + (−0.428 + 0.903i)2-s + (0.589 + 0.807i)3-s + (−0.633 − 0.773i)4-s + (−0.283 + 0.958i)5-s + (−0.982 + 0.186i)6-s + (0.176 + 0.984i)7-s + (0.970 − 0.240i)8-s + (−0.304 + 0.952i)9-s + (−0.745 − 0.666i)10-s + (0.438 − 0.898i)11-s + (0.251 − 0.967i)12-s + (−0.387 − 0.921i)13-s + (−0.964 − 0.262i)14-s + (−0.941 + 0.336i)15-s + (−0.197 + 0.980i)16-s + (−0.997 + 0.0663i)17-s + ⋯
L(s)  = 1  + (−0.428 + 0.903i)2-s + (0.589 + 0.807i)3-s + (−0.633 − 0.773i)4-s + (−0.283 + 0.958i)5-s + (−0.982 + 0.186i)6-s + (0.176 + 0.984i)7-s + (0.970 − 0.240i)8-s + (−0.304 + 0.952i)9-s + (−0.745 − 0.666i)10-s + (0.438 − 0.898i)11-s + (0.251 − 0.967i)12-s + (−0.387 − 0.921i)13-s + (−0.964 − 0.262i)14-s + (−0.941 + 0.336i)15-s + (−0.197 + 0.980i)16-s + (−0.997 + 0.0663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.973 - 0.227i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ 0.973 - 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4072726508 - 0.04702659970i\)
\(L(\frac12)\) \(\approx\) \(0.4072726508 - 0.04702659970i\)
\(L(1)\) \(\approx\) \(0.5740355004 + 0.5502538330i\)
\(L(1)\) \(\approx\) \(0.5740355004 + 0.5502538330i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.428 + 0.903i)T \)
3 \( 1 + (0.589 + 0.807i)T \)
5 \( 1 + (-0.283 + 0.958i)T \)
7 \( 1 + (0.176 + 0.984i)T \)
11 \( 1 + (0.438 - 0.898i)T \)
13 \( 1 + (-0.387 - 0.921i)T \)
17 \( 1 + (-0.997 + 0.0663i)T \)
19 \( 1 + (0.0993 + 0.995i)T \)
23 \( 1 + (0.722 - 0.691i)T \)
29 \( 1 + (0.752 - 0.658i)T \)
31 \( 1 + (-0.294 - 0.955i)T \)
37 \( 1 + (-0.820 - 0.571i)T \)
41 \( 1 + (0.0221 - 0.999i)T \)
43 \( 1 + (-0.903 + 0.428i)T \)
47 \( 1 + (0.999 + 0.0331i)T \)
53 \( 1 + (-0.186 - 0.982i)T \)
59 \( 1 + (0.0331 - 0.999i)T \)
61 \( 1 + (0.650 + 0.759i)T \)
67 \( 1 + (-0.997 - 0.0663i)T \)
71 \( 1 + (-0.970 - 0.240i)T \)
73 \( 1 + (-0.995 + 0.0993i)T \)
79 \( 1 + (-0.930 - 0.367i)T \)
83 \( 1 + (-0.908 - 0.418i)T \)
89 \( 1 + (-0.955 - 0.294i)T \)
97 \( 1 + (0.208 + 0.977i)T \)
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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

−23.38407538029909227238489331717, −22.05429177294830408235660957703, −21.06841142765444097169324888309, −20.24346742000797906273982031853, −19.80654969790981407644320021681, −19.33521612510717701442110839911, −18.05095499117516643581825265891, −17.38897039877946381288024600032, −16.8133477159222727166451109636, −15.50406505382178193308538119067, −14.21238243237659992293015208601, −13.48805732921123080886482969888, −12.832322096592510189049317054344, −11.96972454063125802930991182728, −11.30967904336796749920633281286, −9.99664265020857039978812889483, −9.0291848841533702759892713683, −8.63022310918593453279577271191, −7.27834327602193828317776754953, −7.00132864772351099915238086914, −4.79635061293167202663612888432, −4.23565296203521752829011759973, −3.037989881065609387589670358358, −1.729211951431151419214471927315, −1.16350334836862672517778755214, 0.113200234092746088213884050120, 2.15833111249407621885802298033, 3.2183433537311973602908872631, 4.30546139145997031581979228359, 5.48000498497876269170458517763, 6.21579261596183187719003118673, 7.4264886622372733355067823696, 8.36711594787403289114512326783, 8.87242576918393771767817944062, 9.99248223444301398289711709915, 10.67271698454954986176785965835, 11.60899210035159615622364830476, 13.14037411420917035096375766606, 14.23085097875763012716214918992, 14.724460145622300528008663652466, 15.45074599876742923139517254733, 15.985854292427977044271344142240, 17.07187115296822532569645110129, 17.98604369036836514856618113049, 18.98414873949880038784393163703, 19.28741761353687410625314558704, 20.448363117748834936959930765791, 21.61009301290650116585090628275, 22.407887492403557653742613061518, 22.69982321470702063506063229623

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