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Properties

Label 1-3024-3024.2795-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.249 - 0.968i$
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  + (−0.642 − 0.766i)5-s + (−0.642 + 0.766i)11-s + (−0.342 + 0.939i)13-s − 17-s + i·19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.342 − 0.939i)29-s + (−0.173 − 0.984i)31-s + (−0.866 − 0.5i)37-s + (−0.939 − 0.342i)41-s + (−0.984 − 0.173i)43-s + (0.173 − 0.984i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)5-s + (−0.642 + 0.766i)11-s + (−0.342 + 0.939i)13-s − 17-s + i·19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.342 − 0.939i)29-s + (−0.173 − 0.984i)31-s + (−0.866 − 0.5i)37-s + (−0.939 − 0.342i)41-s + (−0.984 − 0.173i)43-s + (0.173 − 0.984i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.249 - 0.968i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2795, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.249 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5991177098 - 0.4641837362i\)
\(L(\frac12)\) \(\approx\) \(0.5991177098 - 0.4641837362i\)
\(L(1)\) \(\approx\) \(0.7791004219 - 0.06036287999i\)
\(L(1)\) \(\approx\) \(0.7791004219 - 0.06036287999i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.642 - 0.766i)T \)
11 \( 1 + (-0.642 + 0.766i)T \)
13 \( 1 + (-0.342 + 0.939i)T \)
17 \( 1 - T \)
19 \( 1 + iT \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.342 - 0.939i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (-0.984 - 0.173i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.342 + 0.939i)T \)
89 \( 1 + T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−19.299639814601213662327331345417, −18.33222335318053425522113025192, −18.06477445379735929346432657396, −17.13343233569007841020742725809, −16.31728164024357085892915258773, −15.4611448214985585990334275345, −15.241783087831548847376117375950, −14.37550852103431818690071689946, −13.49717554021212544368229520159, −12.97011220790994073637133779819, −12.082247573626498479983585339652, −11.246525284240781369976851354224, −10.72255327189895391741389183207, −10.24907522521953540240499107000, −8.98770240361336101604848144626, −8.48730914477036047077951911724, −7.609313716330874205746525320247, −6.94619160545014598262116615754, −6.31579519131637386378559238395, −5.159283849528888950267056269830, −4.71201319025724138238352614251, −3.32091062051140947726104748679, −3.124190453817593552067498416124, −2.14736231081999442874724988766, −0.74267719161464731001153377043, 0.31268663361090333727204925575, 1.69717037879534096925959785763, 2.26180050447112487157086533299, 3.58256336727698593932449582685, 4.21985337452598380712369059095, 4.942133894726969899815265356942, 5.61744195557400000182624379400, 6.80638497355860620357746315658, 7.338650609405828892909757745913, 8.17124191495593058673470473802, 8.8460359538982542812788447861, 9.580274482790132791655618014912, 10.29652910742195364174201820279, 11.33238670873270591421886494663, 11.79550481990384955421364186296, 12.58572389145284314155997642114, 13.179199688678599880103766528620, 13.88337968552959419931040280686, 14.9626832838774484031096247452, 15.37714264408165354335602288419, 16.06327283388809953193844229921, 17.039117705285957738908921355403, 17.139343249190463653241409786688, 18.39807567389047719470876828440, 18.8630633502343440183135326728

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