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Properties

Label 1-572-572.127-r0-0-0
Degree $1$
Conductor $572$
Sign $0.909 + 0.415i$
Analytic cond. $2.65635$
Root an. cond. $2.65635$
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  + (0.978 + 0.207i)3-s + (0.809 + 0.587i)5-s + (0.978 − 0.207i)7-s + (0.913 + 0.406i)9-s + (0.669 + 0.743i)15-s + (0.104 − 0.994i)17-s + (−0.669 + 0.743i)19-s + 21-s + (0.5 − 0.866i)23-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (−0.669 − 0.743i)29-s + (−0.809 + 0.587i)31-s + (0.913 + 0.406i)35-s + (−0.669 − 0.743i)37-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)3-s + (0.809 + 0.587i)5-s + (0.978 − 0.207i)7-s + (0.913 + 0.406i)9-s + (0.669 + 0.743i)15-s + (0.104 − 0.994i)17-s + (−0.669 + 0.743i)19-s + 21-s + (0.5 − 0.866i)23-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (−0.669 − 0.743i)29-s + (−0.809 + 0.587i)31-s + (0.913 + 0.406i)35-s + (−0.669 − 0.743i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.909 + 0.415i$
Analytic conductor: \(2.65635\)
Root analytic conductor: \(2.65635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 572,\ (0:\ ),\ 0.909 + 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.454530169 + 0.5334902803i\)
\(L(\frac12)\) \(\approx\) \(2.454530169 + 0.5334902803i\)
\(L(1)\) \(\approx\) \(1.770375417 + 0.2423045619i\)
\(L(1)\) \(\approx\) \(1.770375417 + 0.2423045619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (0.978 - 0.207i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.669 - 0.743i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.913 - 0.406i)T \)
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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

−23.63919730990932381896059297263, −21.87538857892377797011215828724, −21.49406947479010196913673635670, −20.6715145080334923742538256687, −20.02617012802839973619072547210, −19.062220336924411488335170372953, −18.16742853181804527958531882204, −17.419211819418910266589809454174, −16.60956800434758681853859436845, −15.20374872676392663889920141491, −14.838212317546705828898810749464, −13.763476592032262866575085866828, −13.16178117099338079998376579023, −12.34869768496869309954897041226, −11.13111344742266613752828282550, −10.10521342263643372987444467917, −9.09406989942123642736858373408, −8.56024696696364020559420426439, −7.66810688074956715676801145379, −6.54880415810842858098252887826, −5.36652089688324498727285929762, −4.49403982638035695045500525900, −3.28491819755340163861748155721, −1.97010104440611547764743408997, −1.4852747046500978587388461137, 1.58313655247508548234471047536, 2.34595923002705648921936032249, 3.412659812119115801721670156082, 4.53303812236875667615079269625, 5.51018543993096978508279064325, 6.837425334743342310658737878163, 7.61224691294017487912672046414, 8.61275919965742564823774527462, 9.41860469667977284496153677917, 10.39698831826595598151169319220, 10.99817242846743706852905187145, 12.34303952241570573082791593624, 13.41122562608415064255610139824, 14.16845482715610619519633616777, 14.60639556664841641264065510740, 15.46905938388751500416778987798, 16.64685965162704948216460911186, 17.5020294920211493921055160010, 18.53292865318254359481042638844, 18.897933742985208359693227568524, 20.29778627272024053849217059629, 20.7607639640236345988737331153, 21.45695838875403958688201117858, 22.2915632615679094644387219181, 23.290690376778887047852113241896

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