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Properties

Label 1-6048-6048.1859-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.852 + 0.523i$
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 + (−0.996 − 0.0871i)13-s + (−0.5 + 0.866i)17-s + (0.258 − 0.965i)19-s + (0.342 − 0.939i)23-s + (−0.984 − 0.173i)25-s + (−0.0871 − 0.996i)29-s + (−0.766 + 0.642i)31-s + (−0.707 + 0.707i)37-s + (−0.642 − 0.766i)41-s + (0.422 − 0.906i)43-s + (−0.766 − 0.642i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.0871 − 0.996i)5-s + (0.996 − 0.0871i)11-s + (−0.996 − 0.0871i)13-s + (−0.5 + 0.866i)17-s + (0.258 − 0.965i)19-s + (0.342 − 0.939i)23-s + (−0.984 − 0.173i)25-s + (−0.0871 − 0.996i)29-s + (−0.766 + 0.642i)31-s + (−0.707 + 0.707i)37-s + (−0.642 − 0.766i)41-s + (0.422 − 0.906i)43-s + (−0.766 − 0.642i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.852 + 0.523i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.852 + 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08117775403 - 0.2873757178i\)
\(L(\frac12)\) \(\approx\) \(-0.08117775403 - 0.2873757178i\)
\(L(1)\) \(\approx\) \(0.8658134570 - 0.2354314765i\)
\(L(1)\) \(\approx\) \(0.8658134570 - 0.2354314765i\)

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 \( 1 + (-0.996 - 0.0871i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.0871 - 0.996i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (0.422 - 0.906i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (0.965 + 0.258i)T \)
59 \( 1 + (-0.819 - 0.573i)T \)
61 \( 1 + (0.0871 + 0.996i)T \)
67 \( 1 + (0.422 + 0.906i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.0871 - 0.996i)T \)
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

−17.97303276374379103023610862321, −17.62710004854266714744410823201, −16.693126701012655575545579134863, −16.28293073486125169134260407471, −15.20482141907839724213680719464, −14.86377942195514638926240975190, −14.15196392492700205118428975076, −13.76718789277243742949884033600, −12.75715530960373940335696264864, −12.091661424313610244693772627650, −11.40885149492944789225259606632, −10.968808644518005949374805270797, −10.00358846685036539287175703314, −9.53665212347110601624152605293, −8.9607090981551689110163398929, −7.807340529413935563113048012306, −7.29922265951866286503117732297, −6.74736247738335144234021791552, −6.002311373428542750033652105334, −5.2250697260630475321484738866, −4.40934150303103865474222967136, −3.5214649217379458548513979655, −3.0271852241590905635871721226, −2.05373137662702822666218481978, −1.42392416067276607829753587340, 0.07455644309211688470682351655, 1.0689954086630489447536458156, 1.89095490768051322435465824898, 2.637904692879030178083393088915, 3.75756662468489310460684204971, 4.327131307387995765791177489256, 5.04639011722758502526034037570, 5.6559359874089068679148670858, 6.62981006929984459948030715090, 7.10010007295817639466193056050, 8.07554952818439796276603048824, 8.88089214444753535462743667207, 9.03099721096191199365290011462, 10.031961102299709649710752501455, 10.56793994285951036640247609402, 11.707231716861088311216523944429, 11.912700988497804128137291548547, 12.804361546325066207416397160044, 13.232341219930891437110663550098, 14.048824036783815425300593564845, 14.72146605157066469238472381540, 15.38112735047228725807947202110, 16.032324555646384961873081698, 16.90269432535520229376104150220, 17.2047297629036055273082976948

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