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Properties

Label 1-6048-6048.4475-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.523 + 0.852i$
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

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

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.523 + 0.852i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (4475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.523 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02999909267 - 0.05361968446i\)
\(L(\frac12)\) \(\approx\) \(0.02999909267 - 0.05361968446i\)
\(L(1)\) \(\approx\) \(0.8055155545 - 0.2260403922i\)
\(L(1)\) \(\approx\) \(0.8055155545 - 0.2260403922i\)

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

−18.23189098255979780159170157291, −17.71341182460864162683778777978, −16.72435077368751310845943958376, −16.10914696151236003482851260803, −15.544617622475626968930854620580, −14.69592181191956036472430893576, −14.46025635241317255318387535959, −13.51108079767620554038280811324, −12.8908017285773895391164919111, −12.34850058167746027088507409079, −11.27309520630222521765895362148, −10.814353496226362679331300791803, −10.41023720256728300676053250983, −9.64908038508008595327151380102, −8.5300527028935484276057472710, −8.26925576204819977135744104799, −7.321398228677763748343162744879, −6.68148512829124879842493773826, −6.061273209142386991096792129547, −5.36304022743904136340577789813, −4.39528422986173594483043592877, −3.61617593246830025161016292910, −3.06113730353181682518933469394, −2.1464830199646386959821630714, −1.48220210495288511679767321744, 0.016661706600432302898695198073, 0.93112349462415786486354407578, 1.81996210976700653463368607015, 2.71224711544437487884714845391, 3.486314053550830824172773688818, 4.45527200505217992223275533822, 4.92107440627329432810938976976, 5.67086716582274548905606513662, 6.34234000433761248456226535195, 7.34082479239205986945012945213, 7.9251647144488950263151042193, 8.61829410804971519818951364722, 9.21300966649459755549968588075, 9.83771911985606871607120507885, 10.72804352533618275312251160709, 11.49124883964020028395826565876, 11.78599206424823178068573349734, 13.0984922942053151169516424982, 13.1768436653956089305464087952, 13.62454291421601767903988793187, 14.80754310433231132331083729992, 15.541920616696223447727596632070, 15.88830949537608572385082341504, 16.51775386882332479606356358132, 17.31561303239731153927680262391

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