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Properties

Label 1-6048-6048.4307-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.411 + 0.911i$
Analytic cond. $28.0867$
Root an. cond. $28.0867$
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  + (−0.0871 + 0.996i)5-s + (−0.996 + 0.0871i)11-s + (−0.422 − 0.906i)13-s + 17-s + (−0.707 − 0.707i)19-s + (0.342 − 0.939i)23-s + (−0.984 − 0.173i)25-s + (−0.906 − 0.422i)29-s + (−0.173 − 0.984i)31-s + (0.258 + 0.965i)37-s + (−0.342 + 0.939i)41-s + (−0.573 − 0.819i)43-s + (−0.173 + 0.984i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.0871 + 0.996i)5-s + (−0.996 + 0.0871i)11-s + (−0.422 − 0.906i)13-s + 17-s + (−0.707 − 0.707i)19-s + (0.342 − 0.939i)23-s + (−0.984 − 0.173i)25-s + (−0.906 − 0.422i)29-s + (−0.173 − 0.984i)31-s + (0.258 + 0.965i)37-s + (−0.342 + 0.939i)41-s + (−0.573 − 0.819i)43-s + (−0.173 + 0.984i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.411 + 0.911i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (4307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.411 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9007763813 + 0.5817850657i\)
\(L(\frac12)\) \(\approx\) \(0.9007763813 + 0.5817850657i\)
\(L(1)\) \(\approx\) \(0.8763753959 + 0.1188742862i\)
\(L(1)\) \(\approx\) \(0.8763753959 + 0.1188742862i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.0871 + 0.996i)T \)
11 \( 1 + (-0.996 + 0.0871i)T \)
13 \( 1 + (-0.422 - 0.906i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.906 - 0.422i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (-0.573 - 0.819i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (-0.906 + 0.422i)T \)
61 \( 1 + (-0.819 + 0.573i)T \)
67 \( 1 + (0.996 + 0.0871i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.906 - 0.422i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−17.4509064817802440417562296410, −16.76570813388424000361632198395, −16.41751402252381361812356266449, −15.74397650827424442485813926650, −14.9896497131688519960278441317, −14.2831310170809733271145735995, −13.61180225805688665047309592771, −12.83931715330839893397402362321, −12.451182348820917177821568424913, −11.7513125644081817118822572300, −11.00511518706565933436208149794, −10.21470351527598432274829040329, −9.5510999441826305383116063942, −8.94059100527989828377492117132, −8.19171951488488556535084876440, −7.6284350806404183951701007476, −6.92034135566755752011560552376, −5.8719861468118372689684490519, −5.28650255560725571863462911802, −4.78330401289533402204830944653, −3.823729331933766971748574285271, −3.25189399377969155848239569512, −1.99435142473813015519722185224, −1.60620413418705109196963409495, −0.377356915209193451592121383056, 0.66043267007327064323035276920, 1.95963768539312038888485226133, 2.78312192234908226100418813328, 3.06978230386408383348196304695, 4.13621816278509081554970859186, 4.91289126674569843032148019417, 5.7061712799206400396142225683, 6.31324813772554661529535603152, 7.17320924454482638975674890148, 7.77024355863693422569937131684, 8.2171210277277955857840552902, 9.30078686499375383471430698717, 10.067578192847233114557863213360, 10.48797363779019235640110478345, 11.10625251891004403063269130061, 11.82809506103979364182345270436, 12.6918514723436776127095840946, 13.16559868279445310013986528686, 13.91625175187139554921519904640, 14.74706238932768101298086886404, 15.20736810317744490818214219440, 15.54983698382854722095264159645, 16.73384522403552751173251641033, 17.042015430044064480602919547267, 18.001176789787668531595271090878

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