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Properties

Label 1-6048-6048.1955-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.629 - 0.776i$
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  + (−0.573 − 0.819i)5-s + (0.819 + 0.573i)11-s + (−0.819 + 0.573i)13-s + (−0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.642 + 0.766i)23-s + (−0.342 + 0.939i)25-s + (0.573 − 0.819i)29-s + (−0.173 + 0.984i)31-s + (−0.707 + 0.707i)37-s + (−0.984 − 0.173i)41-s + (−0.0871 + 0.996i)43-s + (−0.173 − 0.984i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.573 − 0.819i)5-s + (0.819 + 0.573i)11-s + (−0.819 + 0.573i)13-s + (−0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.642 + 0.766i)23-s + (−0.342 + 0.939i)25-s + (0.573 − 0.819i)29-s + (−0.173 + 0.984i)31-s + (−0.707 + 0.707i)37-s + (−0.984 − 0.173i)41-s + (−0.0871 + 0.996i)43-s + (−0.173 − 0.984i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.629 - 0.776i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1955, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.629 - 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8975936279 - 0.4276707089i\)
\(L(\frac12)\) \(\approx\) \(0.8975936279 - 0.4276707089i\)
\(L(1)\) \(\approx\) \(0.8336076777 - 0.08406309762i\)
\(L(1)\) \(\approx\) \(0.8336076777 - 0.08406309762i\)

Euler product

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

−17.77044151482855138903979631576, −17.129419663771314479000656081595, −16.58562181127751077062867655100, −15.66751522996831641269339132143, −15.1402778706652368361068473018, −14.58595177661709582090149930484, −14.048466673977400267804660799741, −13.20747059594597703256108309355, −12.295233549906644480878383006899, −12.05050147614466751345177170281, −10.892581473871545398600247559730, −10.78031662543093191992301496471, −9.97136811782257703609855346336, −9.03716408010641877882087820899, −8.38933888535682089582196952431, −7.793759797515933495697847851313, −6.90743187415692770224382947144, −6.448810196673310993451592260474, −5.76029575566974667097114121753, −4.693832581939318288831467841516, −4.02611214478247373213796440288, −3.41340188557073113605214370155, −2.54977888904823065142927706176, −1.889560164479158369389454402889, −0.59963050419148473623235588311, 0.3877073983396755886300144469, 1.59919619378074991368170574756, 2.07169721769220350619800138192, 3.26982631844192077898561410060, 4.00148624950945616469482201616, 4.72827202035573705450638700760, 5.061181778913139278062690137346, 6.246947619637955886007324531986, 6.88426720184921393369615020810, 7.502276428645667036854179952909, 8.39099274278274640912197922308, 8.84468469359858019784510068290, 9.6935551002895986563464208523, 10.066549025048345206734925545898, 11.28205998657539415303806424289, 11.77414223592735701432427410294, 12.22682195512345854026785304907, 12.90584315289922989105875600279, 13.72371661332294085327503520205, 14.30873757315502441982826227835, 15.12714402456568876882277435041, 15.613487501130378962504022045090, 16.37407979932665264668227321547, 16.95402348656563928476222210378, 17.45451916890666437411948631185

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