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Properties

Label 1-539-539.405-r1-0-0
Degree $1$
Conductor $539$
Sign $-0.446 + 0.894i$
Analytic cond. $57.9235$
Root an. cond. $57.9235$
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  + (0.983 + 0.178i)2-s + (−0.858 − 0.512i)3-s + (0.936 + 0.351i)4-s + (0.393 + 0.919i)5-s + (−0.753 − 0.657i)6-s + (0.858 + 0.512i)8-s + (0.473 + 0.880i)9-s + (0.222 + 0.974i)10-s + (−0.623 − 0.781i)12-s + (−0.983 − 0.178i)13-s + (0.134 − 0.990i)15-s + (0.753 + 0.657i)16-s + (0.963 + 0.266i)17-s + (0.309 + 0.951i)18-s + (−0.309 + 0.951i)19-s + (0.0448 + 0.998i)20-s + ⋯
L(s)  = 1  + (0.983 + 0.178i)2-s + (−0.858 − 0.512i)3-s + (0.936 + 0.351i)4-s + (0.393 + 0.919i)5-s + (−0.753 − 0.657i)6-s + (0.858 + 0.512i)8-s + (0.473 + 0.880i)9-s + (0.222 + 0.974i)10-s + (−0.623 − 0.781i)12-s + (−0.983 − 0.178i)13-s + (0.134 − 0.990i)15-s + (0.753 + 0.657i)16-s + (0.963 + 0.266i)17-s + (0.309 + 0.951i)18-s + (−0.309 + 0.951i)19-s + (0.0448 + 0.998i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.446 + 0.894i$
Analytic conductor: \(57.9235\)
Root analytic conductor: \(57.9235\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (1:\ ),\ -0.446 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346547007 + 2.175969583i\)
\(L(\frac12)\) \(\approx\) \(1.346547007 + 2.175969583i\)
\(L(1)\) \(\approx\) \(1.459199820 + 0.5020306605i\)
\(L(1)\) \(\approx\) \(1.459199820 + 0.5020306605i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.983 + 0.178i)T \)
3 \( 1 + (-0.858 - 0.512i)T \)
5 \( 1 + (0.393 + 0.919i)T \)
13 \( 1 + (-0.983 - 0.178i)T \)
17 \( 1 + (0.963 + 0.266i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-0.550 + 0.834i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.550 + 0.834i)T \)
41 \( 1 + (-0.858 - 0.512i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.134 - 0.990i)T \)
53 \( 1 + (-0.963 + 0.266i)T \)
59 \( 1 + (-0.858 + 0.512i)T \)
61 \( 1 + (0.963 + 0.266i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.0448 + 0.998i)T \)
73 \( 1 + (-0.134 + 0.990i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.983 + 0.178i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−22.90534026485314879670047180002, −22.055003403957530386067375971943, −21.30873599539533332816620700538, −20.876944746070402831176781703007, −19.86109510592470233452475160051, −18.93992992625732506146761270981, −17.41014914014284884711869004555, −17.00200536581965918699153503966, −16.08429351500396584234380697565, −15.35179939481119417519084876750, −14.42273622660806122527187864441, −13.32454941181608402136902899488, −12.60142182193704400377168738489, −11.830235911004847910157663970866, −11.12289210420463559279816592626, −9.89690128874655167150784010262, −9.433617971085440835782890343139, −7.75411207987424502126416715331, −6.640447247292639930621449294765, −5.681036019268857227128884932103, −4.96545435945845735831261484038, −4.358288067275321828865755332925, −3.07174047078718460610834745831, −1.68007999428465602703969638279, −0.478457195252746224300661860224, 1.490500010442689688255623720457, 2.52440930929421296631847316828, 3.60028888697700240959198989473, 4.987180371793838065101342614760, 5.63573750382822237475658872063, 6.64056368054409970874306781611, 7.15744758245205210749509059745, 8.171789104014698429658386961550, 10.18723685093095084697622364432, 10.53018554756237314315300046298, 11.7136976568836464786005483744, 12.32207904487613617433244153829, 13.14872349951827744806892946134, 14.18198013882022190552919061568, 14.73400944877815174432749146555, 15.76585842548452697231651773499, 16.96816045143528120344739142768, 17.188915313112296644738844755290, 18.586087505693663972146679673644, 19.0642327030143094538719999623, 20.32468918091921370516696022966, 21.40839924517112029064351319302, 21.99752521533069378892060885379, 22.74970210376484046774736221151, 23.25826590423104449689545294542

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