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Properties

Label 1-569-569.22-r1-0-0
Degree $1$
Conductor $569$
Sign $0.795 - 0.606i$
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  + (0.616 − 0.787i)2-s + (0.973 + 0.230i)3-s + (−0.240 − 0.970i)4-s + (0.937 − 0.346i)5-s + (0.780 − 0.624i)6-s + (0.262 + 0.964i)7-s + (−0.912 − 0.408i)8-s + (0.894 + 0.448i)9-s + (0.304 − 0.952i)10-s + (0.878 − 0.477i)11-s + (−0.0110 − 0.999i)12-s + (−0.826 + 0.562i)13-s + (0.921 + 0.387i)14-s + (0.992 − 0.121i)15-s + (−0.883 + 0.467i)16-s + (0.773 + 0.633i)17-s + ⋯
L(s)  = 1  + (0.616 − 0.787i)2-s + (0.973 + 0.230i)3-s + (−0.240 − 0.970i)4-s + (0.937 − 0.346i)5-s + (0.780 − 0.624i)6-s + (0.262 + 0.964i)7-s + (−0.912 − 0.408i)8-s + (0.894 + 0.448i)9-s + (0.304 − 0.952i)10-s + (0.878 − 0.477i)11-s + (−0.0110 − 0.999i)12-s + (−0.826 + 0.562i)13-s + (0.921 + 0.387i)14-s + (0.992 − 0.121i)15-s + (−0.883 + 0.467i)16-s + (0.773 + 0.633i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.795 - 0.606i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ 0.795 - 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.264626895 - 1.778477381i\)
\(L(\frac12)\) \(\approx\) \(5.264626895 - 1.778477381i\)
\(L(1)\) \(\approx\) \(2.410641471 - 0.7657642743i\)
\(L(1)\) \(\approx\) \(2.410641471 - 0.7657642743i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.616 - 0.787i)T \)
3 \( 1 + (0.973 + 0.230i)T \)
5 \( 1 + (0.937 - 0.346i)T \)
7 \( 1 + (0.262 + 0.964i)T \)
11 \( 1 + (0.878 - 0.477i)T \)
13 \( 1 + (-0.826 + 0.562i)T \)
17 \( 1 + (0.773 + 0.633i)T \)
19 \( 1 + (0.515 + 0.856i)T \)
23 \( 1 + (0.999 + 0.0331i)T \)
29 \( 1 + (-0.0993 + 0.995i)T \)
31 \( 1 + (-0.998 + 0.0552i)T \)
37 \( 1 + (-0.867 - 0.496i)T \)
41 \( 1 + (0.730 - 0.683i)T \)
43 \( 1 + (0.787 - 0.616i)T \)
47 \( 1 + (-0.336 - 0.941i)T \)
53 \( 1 + (0.624 + 0.780i)T \)
59 \( 1 + (-0.941 + 0.336i)T \)
61 \( 1 + (0.873 + 0.487i)T \)
67 \( 1 + (0.773 - 0.633i)T \)
71 \( 1 + (0.912 - 0.408i)T \)
73 \( 1 + (-0.856 + 0.515i)T \)
79 \( 1 + (-0.219 - 0.975i)T \)
83 \( 1 + (-0.967 - 0.251i)T \)
89 \( 1 + (0.0552 - 0.998i)T \)
97 \( 1 + (-0.997 - 0.0773i)T \)
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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

−23.07830381022136725806800932013, −22.45214727849342569733746408429, −21.48619367247951700089799871599, −20.72956100893263442746489354277, −20.05623493217537708252916946477, −18.951417085753295929516976880641, −17.73027800557174652520285162743, −17.39686554741798230983092347431, −16.39723277908528545956859115085, −15.16007332965012915754759253291, −14.51052362662971176740988464062, −14.020687271679251890097283723985, −13.229851195032382031678814455191, −12.50980929706592708606739944070, −11.21587334316612599296530620619, −9.74080565549634586447981144892, −9.36961456586201766422511507514, −8.02373158319347058504988603139, −7.17328287941748919604579267739, −6.76862759885909587501445308432, −5.36009633426173626267439306890, −4.433120458970960064279954003950, −3.31945726808924717067754812710, −2.49411242248656657830586172860, −1.07538051790717106075653889643, 1.362691766063647149068111527618, 1.98644834473354747539912442059, 3.011400042376675696656859181202, 3.95179966031011211258853912542, 5.155034690717829176143860751208, 5.76942957160782062904544330228, 7.099109746361773818322001027229, 8.73093236204687501588146177373, 9.122843492328575873133143070977, 9.917239601398908311270691440234, 10.878598749832231469808030869351, 12.20545013135285150085400217191, 12.60744825099522324188906166376, 13.77062942708562480231847451379, 14.46269643007734089582721769276, 14.78640317817633653051140439548, 16.08071661290135678912231195549, 17.09241685025529206302591758488, 18.384814433573168749837367201435, 18.97398850537813591862575596168, 19.75562998645973059624047329750, 20.66713733786411980042835559986, 21.44219809177720250532902042063, 21.70618276711857056523470245624, 22.56543947329750426323326175161

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