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Properties

Label 1-6048-6048.3371-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.987 + 0.157i$
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  + (0.996 + 0.0871i)5-s + (0.0871 + 0.996i)11-s + (−0.0871 + 0.996i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (−0.342 + 0.939i)23-s + (0.984 + 0.173i)25-s + (−0.996 + 0.0871i)29-s + (−0.766 + 0.642i)31-s + (−0.707 − 0.707i)37-s + (0.642 + 0.766i)41-s + (0.906 + 0.422i)43-s + (−0.766 − 0.642i)47-s + (−0.258 + 0.965i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.996 + 0.0871i)5-s + (0.0871 + 0.996i)11-s + (−0.0871 + 0.996i)13-s + (−0.5 + 0.866i)17-s + (−0.965 − 0.258i)19-s + (−0.342 + 0.939i)23-s + (0.984 + 0.173i)25-s + (−0.996 + 0.0871i)29-s + (−0.766 + 0.642i)31-s + (−0.707 − 0.707i)37-s + (0.642 + 0.766i)41-s + (0.906 + 0.422i)43-s + (−0.766 − 0.642i)47-s + (−0.258 + 0.965i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.987 + 0.157i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (3371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.987 + 0.157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07597124340 + 0.9601855115i\)
\(L(\frac12)\) \(\approx\) \(0.07597124340 + 0.9601855115i\)
\(L(1)\) \(\approx\) \(1.003964480 + 0.2908576744i\)
\(L(1)\) \(\approx\) \(1.003964480 + 0.2908576744i\)

Euler product

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

−17.36837368164654747112249447865, −16.82488905153317745243115819286, −16.14538297895496455815097793618, −15.50771706590270478548386151748, −14.479159513163752165739698349825, −14.303807620987249725049249463896, −13.258759976394368265523676624920, −13.03010383279620534767168115395, −12.26852690198757338700507247765, −11.21943856691999753769181783397, −10.84026852951494242964915659873, −10.04828743182665546662251449555, −9.473123717394983853983076522500, −8.6274973174631261842157068131, −8.260563051655746332896332518222, −7.17724302319252534253426015306, −6.55067946280010247223230341321, −5.65521005469977042580605306637, −5.48134613095117835610781614303, −4.40272685843085293810111900949, −3.62202740535518021996120438095, −2.64524900665324912429502161764, −2.21856555871419787722643233004, −1.13718162261664752979592235767, −0.21894611846618579962777249466, 1.55586837619883718375822117156, 1.84759091064571851041740673357, 2.57398721878326036675511831992, 3.75587440804235332009315596160, 4.31286227148199498117802711090, 5.17882029835308045152307831172, 5.83407109442865096727988418874, 6.684405691618094844108533690689, 7.00822722324095902672481068227, 7.99719409698246318458252757694, 8.877238786143834738126759666552, 9.43488118696800723626327362649, 9.890434104845059167459319141257, 10.86143250123178395121408535939, 11.17277946573201887435582161188, 12.31705988363505946181287073096, 12.768106765917315184076682590518, 13.36942546951148085873829773306, 14.19107474909312477225175280117, 14.65207253585167166197330348725, 15.2844507874303441264896495205, 16.11745751354226457642352866886, 16.87705960723827005215452321276, 17.40413494840935383943203369205, 17.854982616601485048389543102

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