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Properties

Label 1-579-579.143-r1-0-0
Degree $1$
Conductor $579$
Sign $-0.331 + 0.943i$
Analytic cond. $62.2221$
Root an. cond. $62.2221$
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  + (0.707 − 0.707i)2-s − i·4-s + (0.382 + 0.923i)5-s − 7-s + (−0.707 − 0.707i)8-s + (0.923 + 0.382i)10-s + (−0.382 + 0.923i)11-s + (0.923 − 0.382i)13-s + (−0.707 + 0.707i)14-s − 16-s + (0.382 − 0.923i)17-s + (−0.382 − 0.923i)19-s + (0.923 − 0.382i)20-s + (0.382 + 0.923i)22-s + (0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·4-s + (0.382 + 0.923i)5-s − 7-s + (−0.707 − 0.707i)8-s + (0.923 + 0.382i)10-s + (−0.382 + 0.923i)11-s + (0.923 − 0.382i)13-s + (−0.707 + 0.707i)14-s − 16-s + (0.382 − 0.923i)17-s + (−0.382 − 0.923i)19-s + (0.923 − 0.382i)20-s + (0.382 + 0.923i)22-s + (0.707 − 0.707i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $-0.331 + 0.943i$
Analytic conductor: \(62.2221\)
Root analytic conductor: \(62.2221\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 579,\ (1:\ ),\ -0.331 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3143673228 + 0.4438605461i\)
\(L(\frac12)\) \(\approx\) \(0.3143673228 + 0.4438605461i\)
\(L(1)\) \(\approx\) \(1.134544765 - 0.3080460344i\)
\(L(1)\) \(\approx\) \(1.134544765 - 0.3080460344i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
193 \( 1 \)
good2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 + (0.382 - 0.923i)T \)
19 \( 1 + (-0.382 - 0.923i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.923 + 0.382i)T \)
53 \( 1 + (-0.923 + 0.382i)T \)
59 \( 1 + iT \)
61 \( 1 + (-0.382 + 0.923i)T \)
67 \( 1 + (-0.707 + 0.707i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (-0.923 - 0.382i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−23.01149535837466901995555548378, −21.833223064701262043160739891506, −21.27010641605976021511004776680, −20.615137658875294528608606422084, −19.40494366735093745927450044590, −18.57422026288320640599604918101, −17.32432828272849649270239209725, −16.653218878053428099786237207844, −16.09461968640825988776636972266, −15.34195634396198671748696913446, −14.08368923703261269043759992900, −13.399770926482815340674971694427, −12.81710525872090141013401894965, −12.01733103983694035543985013704, −10.80547199844730612972911425374, −9.57995897633864370664295785803, −8.62443974485180392214919603582, −8.04197786413573915021408744384, −6.60405436597495223754046726119, −5.951495671174894703086684741909, −5.23285042759172089800856647825, −3.90135058818394327658296061402, −3.29574777753607767787406549660, −1.72156932961449434450485229855, −0.096777918916699866427575616405, 1.42796639136728034378256289375, 2.87439183420658041397756666597, 3.068597190909091924500709424249, 4.49889786671676211501530955408, 5.5032340611012107304826279089, 6.5923215401119489527170456085, 7.042817885994169511571876279411, 8.82995200204919857560103702087, 9.84393115884629697191862460435, 10.40916620225323367254649014229, 11.21638224242200279095289818453, 12.30611952003377693038347131210, 13.12804658801397317426659908168, 13.719544723844917269734370288, 14.75578225948138157136985088466, 15.436663514733243963159800949188, 16.27973468803733135478129334264, 17.80291796411505820975782950, 18.37277040667934498664690946537, 19.18377534623151134539963204649, 20.028655225031561342247978411455, 20.83364752173198051724220214220, 21.6784210002255095267329735189, 22.50743604288898143014064803744, 23.02541533608584058373551020460

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