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Properties

Label 1-3024-3024.1325-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.604 + 0.796i$
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  + (−0.342 − 0.939i)5-s + (0.342 − 0.939i)11-s + (−0.342 − 0.939i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.173 − 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.342 + 0.939i)29-s + (−0.939 + 0.342i)31-s − i·37-s + (−0.939 + 0.342i)41-s + (0.984 − 0.173i)43-s + (0.939 + 0.342i)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 + (−0.342 − 0.939i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.173 − 0.984i)23-s + (−0.766 + 0.642i)25-s + (−0.342 + 0.939i)29-s + (−0.939 + 0.342i)31-s − i·37-s + (−0.939 + 0.342i)41-s + (0.984 − 0.173i)43-s + (0.939 + 0.342i)47-s + (−0.866 − 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.604 + 0.796i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.604 + 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5701936175 + 0.2830329144i\)
\(L(\frac12)\) \(\approx\) \(0.5701936175 + 0.2830329144i\)
\(L(1)\) \(\approx\) \(0.8323955266 - 0.2309985059i\)
\(L(1)\) \(\approx\) \(0.8323955266 - 0.2309985059i\)

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 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.342 + 0.939i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
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 \( 1 - T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.342 + 0.939i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−18.91128992581440827197365664050, −18.19850537924520234458005326526, −17.28700220495631947681217543624, −16.85646544786320147171854388493, −15.82819291959572444135222084821, −15.22503582723300545397926629295, −14.57225110418505506520166070726, −14.04700348655939150434343735061, −13.2107766451599091812371498281, −12.19065331065987336170349253103, −11.74469847699208703074361625334, −11.064663233658900242084121314914, −10.140624372526623178665060611385, −9.61999904754952037511168134270, −8.85045757551636173288184910533, −7.6319976727719684983737474358, −7.3509765813784654710676289901, −6.54422716783491382225670230719, −5.78681549511700862539325964468, −4.6876696965368631276083153783, −4.03872875136964645538094111865, −3.229940657206551948533330818147, −2.262875765533462799528846482532, −1.6103293053819193765339264532, −0.134396491352277342960025611296, 0.6760254080319706060406206077, 1.49355766777854072126263006879, 2.6184266471219715095099210474, 3.583097006792226790910734011248, 4.20419453743863149684682543433, 5.21472697211976284873333088169, 5.70392554374263407170551943998, 6.64936140710022517368689227310, 7.5855968992650501589029855314, 8.38302871122388270320321672810, 8.76938729942241362768202738612, 9.58816655479543777498509408247, 10.70435543919533348556343002816, 10.98005503966976689783045702314, 12.13042721656378370928082561006, 12.70229806455302101159084922088, 13.06320853078600717943872417242, 14.18156436693960164441625326159, 14.749509736453659597954483370363, 15.575609169655065576047531403000, 16.249445377792433338123502227091, 16.93236690327559895247469228826, 17.33539599461118537639420561144, 18.335330990556726773585970995198, 19.13089837449416561539911551177

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