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Properties

Label 1-3024-3024.1003-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.700 - 0.713i$
Analytic cond. $324.973$
Root an. cond. $324.973$
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  + (−0.984 + 0.173i)5-s + (−0.984 − 0.173i)11-s + (−0.642 − 0.766i)13-s + 17-s + i·19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (0.642 − 0.766i)29-s + (0.939 + 0.342i)31-s + (−0.866 − 0.5i)37-s + (−0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (0.939 − 0.342i)47-s + (−0.866 − 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)5-s + (−0.984 − 0.173i)11-s + (−0.642 − 0.766i)13-s + 17-s + i·19-s + (0.766 − 0.642i)23-s + (0.939 − 0.342i)25-s + (0.642 − 0.766i)29-s + (0.939 + 0.342i)31-s + (−0.866 − 0.5i)37-s + (−0.766 + 0.642i)41-s + (0.342 + 0.939i)43-s + (0.939 − 0.342i)47-s + (−0.866 − 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.700 - 0.713i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.700 - 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.043671360 - 0.4380283279i\)
\(L(\frac12)\) \(\approx\) \(1.043671360 - 0.4380283279i\)
\(L(1)\) \(\approx\) \(0.8086782423 + 0.01974626222i\)
\(L(1)\) \(\approx\) \(0.8086782423 + 0.01974626222i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.984 + 0.173i)T \)
11 \( 1 + (-0.984 - 0.173i)T \)
13 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + T \)
19 \( 1 + iT \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.642 - 0.766i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.642 - 0.766i)T \)
61 \( 1 + (0.342 + 0.939i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.642 + 0.766i)T \)
89 \( 1 - T \)
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

−19.06816813065320369738478650927, −18.474294377060300012993387699947, −17.403134166263662814701692054687, −16.94031028096891173335923535783, −16.014427742423844332749077633149, −15.52130583750691861693424604584, −14.95666251382896819074968393213, −14.00432687973795866887755823543, −13.388522990015532730832699209733, −12.30532823212223877519817004717, −12.138350215574042829134892547440, −11.16699955001514389352298528073, −10.53748871159070839997877579417, −9.68746381269961481067526455971, −8.85104761012458681754076067512, −8.19810926777834142332306287082, −7.20189236378081978645997908729, −7.11179667258612435037368690468, −5.735069017491301642180167194640, −4.88850246573484306344229684489, −4.46184604561887776989155274730, −3.29112092039485131659921014880, −2.77968279641801784954297683945, −1.57620547974847208941652508862, −0.53887475504367959959801000195, 0.3305500508345211367653080178, 1.22317435219955654328743326027, 2.672571275973574925823009889, 3.06669548753553693599284231877, 4.03409315488290016428379371650, 4.90637382929043093183598178852, 5.54104747244750948388672784476, 6.54227651552407475784377645917, 7.40984360697194658824840564072, 8.06188945817177636540644692724, 8.39191031923463206884500645977, 9.68420534757795180713438782941, 10.358416911095380589172667621959, 10.852597295420291287409983093653, 11.89866141950836673694012593181, 12.34034019320916775583817706338, 12.99758457241949702682202928132, 13.99270383610301236286547747774, 14.686725570308360489137321190091, 15.32683563414693817443788518980, 15.93408465514141590267098231360, 16.612155432493922875060585212206, 17.36328802642551067148316037768, 18.24227096071152043269492101140, 18.88516169240659409514356914532

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