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Properties

Label 1-6039-6039.524-r0-0-0
Degree $1$
Conductor $6039$
Sign $-0.768 + 0.639i$
Analytic cond. $28.0449$
Root an. cond. $28.0449$
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)2-s + (0.309 + 0.951i)4-s + (0.104 − 0.994i)5-s + 7-s + (−0.309 + 0.951i)8-s + (0.669 − 0.743i)10-s + (−0.669 + 0.743i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.104 − 0.994i)17-s + (0.5 + 0.866i)19-s + (0.978 − 0.207i)20-s + (0.669 + 0.743i)23-s + (−0.978 − 0.207i)25-s + (−0.978 + 0.207i)26-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.104 − 0.994i)5-s + 7-s + (−0.309 + 0.951i)8-s + (0.669 − 0.743i)10-s + (−0.669 + 0.743i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.104 − 0.994i)17-s + (0.5 + 0.866i)19-s + (0.978 − 0.207i)20-s + (0.669 + 0.743i)23-s + (−0.978 − 0.207i)25-s + (−0.978 + 0.207i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $-0.768 + 0.639i$
Analytic conductor: \(28.0449\)
Root analytic conductor: \(28.0449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6039} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6039,\ (0:\ ),\ -0.768 + 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8334586603 + 2.306003960i\)
\(L(\frac12)\) \(\approx\) \(0.8334586603 + 2.306003960i\)
\(L(1)\) \(\approx\) \(1.488430733 + 0.7138068937i\)
\(L(1)\) \(\approx\) \(1.488430733 + 0.7138068937i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.104 - 0.994i)T \)
7 \( 1 + T \)
13 \( 1 + (-0.669 + 0.743i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.669 + 0.743i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (0.978 - 0.207i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.669 + 0.743i)T \)
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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

−17.38774322300506312283047935737, −17.1038319690072995472908888687, −15.62963022675072724334905546732, −15.35597902867750012893466986261, −14.61187207441864937134720569727, −14.35978307163240009602352681414, −13.46939830644651713558711155182, −12.95316864589012784980044216739, −12.08384219841269941017278501962, −11.459855035709841907114584719027, −10.98510441525496836222869308697, −10.282842091944371131110303101, −9.88145238664794265947849322123, −8.823616948730041438306490568064, −7.95024901162183601387805219172, −7.221753525110904769028186206229, −6.58029084829675692635452141633, −5.750737862018998875122590146815, −5.1480026280743989672553042044, −4.45061097448659859996833844345, −3.624706762142426943701891991852, −2.911486892876502231691204726169, −2.23730730532461542490000124074, −1.595614444751074342084451158814, −0.40176513051241532580550044582, 1.347218368615618692488464092959, 1.815379603022500260307236740206, 2.95090343338219050987387274665, 3.686708986130771488427998257661, 4.70932004824436499669477421921, 4.95673549791742670036117193640, 5.463357574749720250028808216394, 6.44769415035652009963969589766, 7.32231293211996555377675513704, 7.700823676488832199699075727632, 8.57075547099367178522127458988, 9.07583080833683611029402675357, 9.84496862069385708694350687837, 11.005237747566241509861306086797, 11.57205054426382986201386513509, 12.281656802312799669704992084465, 12.543839923956366423829817558881, 13.65689739451528092416713203184, 13.99699276525773096611925436172, 14.542192882158376944437316716653, 15.353913540870337299089493213875, 16.02550237786317047026005863309, 16.60190691380733950020329337391, 17.075227849410453422506978004282, 17.79012220656854636813308770869

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