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Properties

Label 1-6048-6048.2867-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.476 + 0.879i$
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  + (−0.819 − 0.573i)5-s + (0.573 + 0.819i)11-s + (−0.573 + 0.819i)13-s + (−0.5 + 0.866i)17-s + (−0.258 + 0.965i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (0.819 − 0.573i)29-s + (−0.173 − 0.984i)31-s + (0.707 − 0.707i)37-s + (0.984 − 0.173i)41-s + (0.996 − 0.0871i)43-s + (−0.173 + 0.984i)47-s + (−0.965 − 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.819 − 0.573i)5-s + (0.573 + 0.819i)11-s + (−0.573 + 0.819i)13-s + (−0.5 + 0.866i)17-s + (−0.258 + 0.965i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (0.819 − 0.573i)29-s + (−0.173 − 0.984i)31-s + (0.707 − 0.707i)37-s + (0.984 − 0.173i)41-s + (0.996 − 0.0871i)43-s + (−0.173 + 0.984i)47-s + (−0.965 − 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.476 + 0.879i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.476 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5230594103 + 0.8783498248i\)
\(L(\frac12)\) \(\approx\) \(0.5230594103 + 0.8783498248i\)
\(L(1)\) \(\approx\) \(0.8686587841 + 0.1455992774i\)
\(L(1)\) \(\approx\) \(0.8686587841 + 0.1455992774i\)

Euler product

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

−17.62365875127099690141712152165, −16.736595538377858395848259040490, −16.11940604149535193533090218300, −15.52001768302444312339904392799, −14.90342740016027641158298552508, −14.27566020258673772314528368492, −13.67625091401191917466382256210, −12.77206568518320556910724004769, −12.20714398954022491401590207181, −11.46724063540584815465972076082, −10.86168123348786448982221526155, −10.48115058502889512194404855136, −9.33394979632151548635139726456, −8.86265122395073552793723270937, −8.03523211030303938120786225941, −7.43801864000822197591486291248, −6.63927333851483430392097315545, −6.26361260659119496513699960727, −4.94484078318117508844832343107, −4.69360390064436277976759978423, −3.59594487305042426027266595773, −2.9482675145010503430226574147, −2.492229038956619259534144817699, −1.04536815119688355870374901273, −0.31889069973610771469123328144, 1.04364221058951045532707132929, 1.79606201204823132914770062515, 2.61993907889527612903820338662, 3.80367515032380027581609575601, 4.23175188439125942354915977569, 4.729131712150514358510429078400, 5.799840962286155953251897844642, 6.436293610266890585827129756630, 7.39120727410481437296021313783, 7.73442559166019011577160289682, 8.59895164974088256695690444545, 9.3438440316508254460731099443, 9.713165763467704730360455442274, 10.83077504309215749594657588760, 11.33487259338451397127721846938, 12.16020427454399403944646213308, 12.4894625353236460316076983305, 13.16787568985157278627993987355, 14.11872153247684506319277730707, 14.76208594854370877689709345233, 15.23613875261380947313027138460, 16.012776366316741533123144880162, 16.58557495852542325873747520480, 17.352318450021469296910065644454, 17.59000871022422224131333331264

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