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Properties

Label 1-571-571.194-r1-0-0
Degree $1$
Conductor $571$
Sign $0.996 + 0.0803i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (−0.245 + 0.969i)2-s + (−0.980 + 0.197i)3-s + (−0.879 − 0.475i)4-s + (−0.461 − 0.887i)5-s + (0.0495 − 0.998i)6-s + (0.277 + 0.960i)7-s + (0.677 − 0.735i)8-s + (0.922 − 0.386i)9-s + (0.973 − 0.229i)10-s + (−0.973 − 0.229i)11-s + (0.956 + 0.293i)12-s + (0.746 − 0.665i)13-s + (−0.999 + 0.0330i)14-s + (0.627 + 0.778i)15-s + (0.546 + 0.837i)16-s + (−0.965 + 0.261i)17-s + ⋯
L(s)  = 1  + (−0.245 + 0.969i)2-s + (−0.980 + 0.197i)3-s + (−0.879 − 0.475i)4-s + (−0.461 − 0.887i)5-s + (0.0495 − 0.998i)6-s + (0.277 + 0.960i)7-s + (0.677 − 0.735i)8-s + (0.922 − 0.386i)9-s + (0.973 − 0.229i)10-s + (−0.973 − 0.229i)11-s + (0.956 + 0.293i)12-s + (0.746 − 0.665i)13-s + (−0.999 + 0.0330i)14-s + (0.627 + 0.778i)15-s + (0.546 + 0.837i)16-s + (−0.965 + 0.261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.996 + 0.0803i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.996 + 0.0803i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6204824185 + 0.02496783519i\)
\(L(\frac12)\) \(\approx\) \(0.6204824185 + 0.02496783519i\)
\(L(1)\) \(\approx\) \(0.5222594753 + 0.2040941009i\)
\(L(1)\) \(\approx\) \(0.5222594753 + 0.2040941009i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.245 + 0.969i)T \)
3 \( 1 + (-0.980 + 0.197i)T \)
5 \( 1 + (-0.461 - 0.887i)T \)
7 \( 1 + (0.277 + 0.960i)T \)
11 \( 1 + (-0.973 - 0.229i)T \)
13 \( 1 + (0.746 - 0.665i)T \)
17 \( 1 + (-0.965 + 0.261i)T \)
19 \( 1 + (0.909 + 0.416i)T \)
23 \( 1 + (0.115 + 0.993i)T \)
29 \( 1 + (-0.401 - 0.915i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (-0.995 - 0.0990i)T \)
41 \( 1 + (0.986 - 0.164i)T \)
43 \( 1 + (-0.518 - 0.854i)T \)
47 \( 1 + (-0.546 - 0.837i)T \)
53 \( 1 + (-0.997 - 0.0660i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (-0.768 + 0.639i)T \)
67 \( 1 + (-0.997 - 0.0660i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.995 + 0.0990i)T \)
79 \( 1 + (-0.991 - 0.131i)T \)
83 \( 1 + (0.0495 - 0.998i)T \)
89 \( 1 + (-0.0495 + 0.998i)T \)
97 \( 1 + (0.180 - 0.983i)T \)
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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

−22.707930526249688506807488753137, −22.478404355145951388283518496471, −21.31293519827706107687986196753, −20.58265731700082309997059894366, −19.65400127773087067543152842541, −18.662026383755400417294520054910, −18.16852475706160006832305691820, −17.531436212749390846019727683177, −16.449166758998710731847185363536, −15.68404361015234869099290127084, −14.259324797610386879077529071777, −13.46964037014510098059087854160, −12.68585297671211137719106047164, −11.505965296309831451440707775507, −11.05538130874767428242146814081, −10.52707201277387330562301151950, −9.5566681399101992961062554310, −8.102081157920828360236884238215, −7.311361245344715548504747518499, −6.45528524413614296096430763137, −4.91183290620870002674523343615, −4.26900832993846063575508715150, −3.1131865389157594526483858916, −1.87964509730956451524494569539, −0.64074212643700133172035477208, 0.33867175896998771172017655548, 1.5182576821587445283918031359, 3.62404056570329303428023691640, 4.81414656549812621815016871282, 5.44047422592372506100402897302, 6.00588918338841444799029766374, 7.35266549796377923213887455850, 8.22970204171046312262399606306, 8.967419731034360866279595080957, 9.995328852140772189063934699086, 11.03148065445663888448147766300, 11.9694058877408062964132875979, 12.91138818083878354327384057602, 13.58011649467976439112822026716, 15.209059490941502277036085491510, 15.72092778064878023997816510499, 16.04940990159385479109756264740, 17.205856946807050698908344973857, 17.86982502961605238830685825936, 18.49693097772198286523970078860, 19.47008422955070769623899205609, 20.78580050204081412355659189255, 21.49252595077891684454919781213, 22.56039397144961665976953238518, 23.14179768803118168167087633535

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