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Properties

Label 1-572-572.51-r0-0-0
Degree $1$
Conductor $572$
Sign $0.0694 + 0.997i$
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  + (0.809 + 0.587i)3-s + (−0.309 + 0.951i)5-s + (0.809 − 0.587i)7-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)15-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + 21-s − 23-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.309 + 0.951i)31-s + (0.309 + 0.951i)35-s + (0.809 − 0.587i)37-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (−0.309 + 0.951i)5-s + (0.809 − 0.587i)7-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)15-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + 21-s − 23-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.309 + 0.951i)31-s + (0.309 + 0.951i)35-s + (0.809 − 0.587i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.0694 + 0.997i$
Analytic conductor: \(2.65635\)
Root analytic conductor: \(2.65635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 572,\ (0:\ ),\ 0.0694 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.356524893 + 1.265334560i\)
\(L(\frac12)\) \(\approx\) \(1.356524893 + 1.265334560i\)
\(L(1)\) \(\approx\) \(1.294315003 + 0.5546578576i\)
\(L(1)\) \(\approx\) \(1.294315003 + 0.5546578576i\)

Euler product

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

−23.34798005013734116451026915989, −22.079260087987654907168158522000, −21.17742563614021842510673337272, −20.31644146468114561935879421668, −19.99905853040600278407913498991, −18.85915649664091617611264908363, −18.102430072074266784439090982665, −17.4130439302200726141399129330, −16.09164659555629255301048565812, −15.500206779060221824629825849742, −14.477763649211254626677406618231, −13.705388618204092815979751249690, −12.88449177965983278111875477597, −11.9306172352115212709236813330, −11.45204527446609134764414507195, −9.72652045270812585396902004582, −9.053785609521595336236100507671, −8.17047084681039705077455443849, −7.6515691791193102498722337165, −6.40093186393058545897550960349, −5.15627084884413090582057557739, −4.356041017570026254648906260925, −3.03028727958666240035855711194, −1.99693562084571979622871791296, −0.92791649335905777630050714198, 1.63946164513814431625852195749, 2.73226516877710959234964708565, 3.79687080306175864638893275779, 4.40008431415989954086708356670, 5.75166688506203342136357953992, 7.0389575374000153146000428538, 7.878383023465366044336440647686, 8.489194242815655608551462177321, 9.86233824825013552000394850778, 10.461190870146481084696498529939, 11.21330319886461654403466409269, 12.28527689309448756646862125522, 13.78677597138701745420358684010, 14.085393040606872729380941456176, 14.97909013783949445352960294356, 15.64008330217619536516904437777, 16.609247692578009608774150475701, 17.72121147074862587723795297104, 18.45017437803494752829179675474, 19.55739649693178388654437512792, 19.97261124214631511371042177760, 21.03605833230743426764757976042, 21.65373421878156558729015587379, 22.51510593104366720969078251735, 23.40395201059643138234691547104

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