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Properties

Label 1-6048-6048.5549-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.419 + 0.907i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.419 + 0.907i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.419 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2863371239 + 0.4479143130i\)
\(L(\frac12)\) \(\approx\) \(0.2863371239 + 0.4479143130i\)
\(L(1)\) \(\approx\) \(0.9248157975 - 0.08356533038i\)
\(L(1)\) \(\approx\) \(0.9248157975 - 0.08356533038i\)

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

−17.68975590897868848439410534320, −16.85257201579304454003097012820, −16.19723635404284061860790852496, −15.41126280015984500993250748616, −14.74999172350268678797960332883, −14.42417890479809319533413031136, −13.63245074629637174328367195141, −12.86238624869808515818679087736, −12.29112375626298972406521712485, −11.37778278241699860581184032774, −10.866538286250863420121377305085, −10.332930259499148656537432257430, −9.59035713619673036593341119653, −8.73639972730766812457710120057, −8.22102999673428712212716337007, −7.21853797040307219883565228484, −6.74361542406225231360431178621, −6.06547233181381443952974761660, −5.45100751219621219880001247878, −4.2528918333166091174077817972, −3.81363103761872208969861030361, −2.9829770792174614476897926539, −2.18603857041829887880520145034, −1.4608715658604401195134914933, −0.13040050784808052436431060782, 1.10593277065784450434253434621, 1.75069202349101855730891382424, 2.51596383629025150425560493829, 3.78811922904889658655147635572, 4.246276142944971976827822451051, 4.79314774036105167704539175997, 5.93282025355382006756695813157, 6.23785231871812641935610130563, 7.17975715717073210328329735290, 7.91856589908843243138382784469, 8.82724879168489065320230625277, 9.17126572703581184996947730899, 9.64091121080719781842451993969, 10.79116482699752206538327326524, 11.48965404351940348918564131731, 11.813710328711754968772628233374, 12.790558650269962731215514255183, 13.31105468947757582675440231178, 13.83383651672444139087612646084, 14.63279090541997019097344411345, 15.43949980688051020443398409174, 15.93948687584051017879058586952, 16.72129641417340513202655075815, 17.23495110072809465199863599014, 17.62715318673022335919492989702

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