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Properties

Label 1-504-504.139-r0-0-0
Degree $1$
Conductor $504$
Sign $-0.939 + 0.342i$
Analytic cond. $2.34056$
Root an. cond. $2.34056$
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  + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s − 17-s − 19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s − 37-s + (0.5 + 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s − 53-s + 55-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s − 17-s − 19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s − 37-s + (0.5 + 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s − 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(2.34056\)
Root analytic conductor: \(2.34056\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (0:\ ),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004071089961 + 0.02308829848i\)
\(L(\frac12)\) \(\approx\) \(0.004071089961 + 0.02308829848i\)
\(L(1)\) \(\approx\) \(0.6724863926 - 0.08618080720i\)
\(L(1)\) \(\approx\) \(0.6724863926 - 0.08618080720i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−23.87995965414285862835515229700, −23.43226256430902211040034124998, −22.25743523858755909954561132778, −21.77130949523804883349697669509, −20.81162226786707234221522058358, −19.63499233757252549607847433403, −19.060587742187934420253504346993, −18.33554448201533513186936031975, −17.33927439363880645252618689459, −16.315955741956241078876361782067, −15.57349324962131349747153231801, −14.60691458232948800462697084940, −13.96422991235652961863984972477, −12.85046138296134268010143908288, −11.87972901068914974651610209516, −10.87804191424727105437557890040, −10.474141003438399117447595813237, −8.987931590114394843362473135878, −8.29893565257128632357201495165, −6.98564146429053208506825801387, −6.55777931244056129189787293662, −5.14199384071444302935767765680, −4.07465487987397732496242287955, −3.02464675026313829349321470439, −2.03164970095574523273127762401, 0.01184424389964870502331897359, 1.653922853883589027838396516888, 2.875959402880781511755916234770, 4.29521240203592245927285510607, 4.88229338306133518620639603201, 6.03583264032524086954109844505, 7.34810552647647487933671341746, 8.03836891032418533163426393033, 9.03915238459091840725191919566, 9.943203633622653490448844297239, 10.99471059869992019814439899701, 11.96606799887637021270589087790, 12.893340195457389930848241239351, 13.30528330599612662988811153781, 14.918859073053944247391753320600, 15.35779019969525116023984392837, 16.28020371827729638291515211197, 17.36214245516329706741787764277, 17.79061082800963494770141675973, 19.19153348543663365595629679663, 19.78958921567821776615930230612, 20.60976333361904519054831335652, 21.30499504659293794386969795965, 22.49229253393376060758809071575, 23.19053062826645176296163836417

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