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Properties

Label 1-3024-3024.1787-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.713 + 0.700i$
Analytic cond. $14.0433$
Root an. cond. $14.0433$
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  + (−0.342 + 0.939i)5-s + (−0.342 − 0.939i)11-s + (0.984 − 0.173i)13-s − 17-s + i·19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.984 + 0.173i)29-s + (−0.766 + 0.642i)31-s + (−0.866 − 0.5i)37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (0.766 + 0.642i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)5-s + (−0.342 − 0.939i)11-s + (0.984 − 0.173i)13-s − 17-s + i·19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.984 + 0.173i)29-s + (−0.766 + 0.642i)31-s + (−0.866 − 0.5i)37-s + (0.173 + 0.984i)41-s + (0.642 − 0.766i)43-s + (0.766 + 0.642i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.713 + 0.700i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.713 + 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.274487227 + 0.5209451791i\)
\(L(\frac12)\) \(\approx\) \(1.274487227 + 0.5209451791i\)
\(L(1)\) \(\approx\) \(0.9757747502 + 0.1430996994i\)
\(L(1)\) \(\approx\) \(0.9757747502 + 0.1430996994i\)

Euler product

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

−19.103057056777516148780901254147, −18.052099788958659625725288767098, −17.63678248527932076594403853393, −16.926914342541479903396511909937, −15.95955443110490827046977400160, −15.61229737856896935480745829288, −15.03247188208751695729656403152, −13.75026179476744719007582768030, −13.38787390688259655695182219443, −12.65534941616745795691103830809, −11.922991204049337318731617775729, −11.24976655796997234902391914409, −10.49968473988160989358015853548, −9.49481566923836032333117961397, −8.96326985209657350739803875178, −8.29570084688944803578645401930, −7.43187696825749838730495692163, −6.76338233865582728933991504353, −5.753771881817633639661615938112, −4.99039535068420998646173791969, −4.30713595704636369874590678779, −3.63614306101388693540907716621, −2.40753571052105476628836009837, −1.642483650479300643999515199670, −0.5834335856632185278563772064, 0.766454667718673501014738515071, 2.00930621465828682520950366610, 2.8693107188402087570109778564, 3.59415124130865128234729932286, 4.25961513087454345478109623785, 5.42377986980462037697424467302, 6.20949426427461134017003367226, 6.70498928255576319000582730831, 7.67236558095161563381494081371, 8.39377354906802168234734194446, 8.93491880355760477357084739083, 10.12929293694945865692962986341, 10.84438568887752885476723609617, 11.01133454971332577964021568264, 12.08278078531949038865135694133, 12.74718599096785590719256089631, 13.795559062537874167442194486180, 14.08362025964208747544564419060, 14.934251014247963514579485283697, 15.89710440565084791351466302746, 15.98990078351681299501724097651, 17.06634955442915168474222128292, 17.980970801095981363322070833731, 18.47752538483117337182855368443, 18.94905821016371526344579873120

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