Properties

Label 1-6048-6048.691-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.0322 - 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0322 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0322 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.0322 - 0.999i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.0322 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6704527142 - 0.6924046115i\)
\(L(\frac12)\) \(\approx\) \(0.6704527142 - 0.6924046115i\)
\(L(1)\) \(\approx\) \(1.013899300 + 0.05504286086i\)
\(L(1)\) \(\approx\) \(1.013899300 + 0.05504286086i\)

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

−17.52271357796138926027582682595, −17.2818067789302882894136204848, −16.752582731139583513637285772833, −15.960716032697497356190264492435, −15.230188904120424129965666511641, −14.59086904592371935615503635450, −13.90390680771535900871896135397, −13.21338937372390958584467461786, −12.60503631995039578200270426333, −12.17394791810690228338583162208, −11.1541076462969587919249699281, −10.663561086036483890488472646353, −9.73493730239294109551614994385, −9.27854332493287615000036477695, −8.461773175728569815428681258999, −8.12271276189175363711366543959, −6.94576876831255770418548790499, −6.36877342128339927282950726918, −5.70579507557748078141169369370, −4.83057884186590935856747660212, −4.43584795014177221667049892976, −3.354121836981888251460276171309, −2.667093198550958661025186800222, −1.58107764434261012802682478107, −1.14104985553449637331537818015, 0.22843889555319669232269067734, 1.59030671854353950060123722988, 2.32697523656638899493790539722, 2.752343890056975730124153662601, 3.96130049998269524763513544326, 4.44247467899739750880828917782, 5.32418894805155430652440385822, 6.21537638718219410745845969290, 6.8266347166521336846978918466, 7.162015147537529272121548723906, 8.15928242664426842323359698277, 9.10289230435241264810383196206, 9.51189881675906386782780640342, 10.2607717852425054793987310753, 10.813204770816256196371034596876, 11.770292121240288536170421995329, 12.025811773611400669573874466772, 13.104980339268117012396568456746, 13.620792081258055052861085824147, 14.372118985287576580382624800103, 14.91547609615718842570049364816, 15.265074515486878993078303725382, 16.44783681579602560386779021017, 16.928421789783408548198011574764, 17.531147297870229562377533479753

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