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Properties

Label 24-3248e12-1.1-c0e12-0-3
Degree $24$
Conductor $1.378\times 10^{42}$
Sign $1$
Analytic cond. $329.062$
Root an. cond. $1.27317$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4-s − 2·7-s + 2·9-s − 4·14-s + 4·18-s − 2·28-s − 2·29-s + 2·36-s − 4·43-s + 49-s − 2·53-s − 4·58-s − 4·63-s − 2·67-s − 12·79-s + 81-s − 8·86-s + 2·98-s − 4·106-s + 2·107-s − 2·109-s + 2·113-s − 2·116-s − 5·121-s − 8·126-s + 127-s + ⋯
L(s)  = 1  + 2·2-s + 4-s − 2·7-s + 2·9-s − 4·14-s + 4·18-s − 2·28-s − 2·29-s + 2·36-s − 4·43-s + 49-s − 2·53-s − 4·58-s − 4·63-s − 2·67-s − 12·79-s + 81-s − 8·86-s + 2·98-s − 4·106-s + 2·107-s − 2·109-s + 2·113-s − 2·116-s − 5·121-s − 8·126-s + 127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 7^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(329.062\)
Root analytic conductor: \(1.27317\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 7^{12} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9187158585\)
\(L(\frac12)\) \(\approx\) \(0.9187158585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
7 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
good3 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
5 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
11 \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
13 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
17 \( ( 1 + T^{4} )^{6} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
23 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
31 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
41 \( ( 1 + T^{4} )^{6} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \)
47 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
59 \( ( 1 + T^{4} )^{6} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
73 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
79 \( ( 1 + T )^{12}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
89 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
97 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.99599150712857934941975770975, −2.96642885580976668833550417873, −2.90351166938755651144905297906, −2.73396214808628871645455393542, −2.48483352522857650389283157982, −2.44720181430312701323687841018, −2.32436560840621680276068853852, −2.27275103590952278915121683865, −2.19839818101697887045045046020, −2.05846576136210007949601661612, −1.92141718426481951010729312967, −1.88226202474570688424354551503, −1.87432723604898567349223508544, −1.74291324757815386648191074569, −1.58437005044490372033492446984, −1.50252191565474728465265066436, −1.41089130116961254677077454335, −1.36883316249414105390877704973, −1.34298662864208781908689187042, −1.14805958862100758529039845766, −1.11764691974148437954694957138, −0.962047704168188146380143389449, −0.55640568988175825485423691521, −0.34124707707942638013141617663, −0.21665812015984317353191543278, 0.21665812015984317353191543278, 0.34124707707942638013141617663, 0.55640568988175825485423691521, 0.962047704168188146380143389449, 1.11764691974148437954694957138, 1.14805958862100758529039845766, 1.34298662864208781908689187042, 1.36883316249414105390877704973, 1.41089130116961254677077454335, 1.50252191565474728465265066436, 1.58437005044490372033492446984, 1.74291324757815386648191074569, 1.87432723604898567349223508544, 1.88226202474570688424354551503, 1.92141718426481951010729312967, 2.05846576136210007949601661612, 2.19839818101697887045045046020, 2.27275103590952278915121683865, 2.32436560840621680276068853852, 2.44720181430312701323687841018, 2.48483352522857650389283157982, 2.73396214808628871645455393542, 2.90351166938755651144905297906, 2.96642885580976668833550417873, 2.99599150712857934941975770975

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

Plot not available for L-functions of degree greater than 10.