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Properties

Label 24-2352e12-1.1-c0e12-0-0
Degree $24$
Conductor $2.866\times 10^{40}$
Sign $1$
Analytic cond. $6.84121$
Root an. cond. $1.08342$
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  − 3-s − 7-s + 9-s + 2·13-s + 19-s + 21-s + 25-s + 31-s − 8·37-s − 2·39-s − 2·43-s + 49-s − 57-s − 5·61-s − 63-s + 67-s − 73-s − 75-s + 79-s − 2·91-s − 93-s − 4·97-s + 103-s − 109-s + 8·111-s + 2·117-s + 121-s + ⋯
L(s)  = 1  − 3-s − 7-s + 9-s + 2·13-s + 19-s + 21-s + 25-s + 31-s − 8·37-s − 2·39-s − 2·43-s + 49-s − 57-s − 5·61-s − 63-s + 67-s − 73-s − 75-s + 79-s − 2·91-s − 93-s − 4·97-s + 103-s − 109-s + 8·111-s + 2·117-s + 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 3^{12} \cdot 7^{24}\)
Sign: $1$
Analytic conductor: \(6.84121\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 3^{12} \cdot 7^{24} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5809011736\)
\(L(\frac12)\) \(\approx\) \(0.5809011736\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
7 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
good5 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
11 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
13 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2} \)
17 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
23 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
37 \( ( 1 + T + T^{2} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
43 \( ( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2} \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
53 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
59 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
61 \( ( 1 + T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
89 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \)
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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

−2.98986270719323247350820099690, −2.96370648453313589758515948768, −2.90812956338175177005206522851, −2.86056993384772504521251370914, −2.81230088966037015923332730853, −2.48566991054796659615350535615, −2.40479910484953922330871911703, −2.40267044638405866376129900135, −2.30763411432016953646803804732, −2.23565466144448216190233748107, −1.98359248044605318829658378785, −1.86285574829364043728374985087, −1.84097077124193558520606011269, −1.75857494847566921037715745747, −1.69627916949802787486959036657, −1.53535039062202461880369649963, −1.45222423205320220948568015029, −1.44578204741789146594300336549, −1.36947168206067206064948540954, −1.20157051551374921051934288539, −1.14815439704503701051685649368, −0.826980987394034488163265169230, −0.69787180881732726523699471557, −0.56987836057923689590271086419, −0.27983509071363694380779464788, 0.27983509071363694380779464788, 0.56987836057923689590271086419, 0.69787180881732726523699471557, 0.826980987394034488163265169230, 1.14815439704503701051685649368, 1.20157051551374921051934288539, 1.36947168206067206064948540954, 1.44578204741789146594300336549, 1.45222423205320220948568015029, 1.53535039062202461880369649963, 1.69627916949802787486959036657, 1.75857494847566921037715745747, 1.84097077124193558520606011269, 1.86285574829364043728374985087, 1.98359248044605318829658378785, 2.23565466144448216190233748107, 2.30763411432016953646803804732, 2.40267044638405866376129900135, 2.40479910484953922330871911703, 2.48566991054796659615350535615, 2.81230088966037015923332730853, 2.86056993384772504521251370914, 2.90812956338175177005206522851, 2.96370648453313589758515948768, 2.98986270719323247350820099690

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

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