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Properties

Label 24-588e12-1.1-c0e12-0-0
Degree $24$
Conductor $1.708\times 10^{33}$
Sign $1$
Analytic cond. $4.07768\times 10^{-7}$
Root an. cond. $0.541710$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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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

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

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{12} \cdot 7^{24}\)
Sign: $1$
Analytic conductor: \(4.07768\times 10^{-7}\)
Root analytic conductor: \(0.541710\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{12} \cdot 7^{24} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3910274449\)
\(L(\frac12)\) \(\approx\) \(0.3910274449\)
\(L(1)\) 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

−3.69269407874133238887112002482, −3.58989539608701462091752793780, −3.57514782796243959163573400822, −3.46481160600584306661912146577, −3.41845638561531586576518534399, −3.38449370565176405325727462572, −3.28239734147959649140593305471, −2.99851932164108026702505154775, −2.97447454796092782650989884802, −2.95331654866925326455233834708, −2.79054051921709119995551940344, −2.69671423820908542755360543616, −2.67402538236873830691499900714, −2.26525933591875023635767300839, −2.26339410567486130003182245167, −2.14655873873253757769721274675, −2.06715429261956956760919668483, −1.81313514156178756525041020053, −1.67903677522444305254266723655, −1.55666845176746133258904324933, −1.54830538513620671623865839789, −1.51278655266175496389788403266, −1.38529926558741575824058485651, −1.23546709850735635284275309449, −0.850387021171205720859953640240, 0.850387021171205720859953640240, 1.23546709850735635284275309449, 1.38529926558741575824058485651, 1.51278655266175496389788403266, 1.54830538513620671623865839789, 1.55666845176746133258904324933, 1.67903677522444305254266723655, 1.81313514156178756525041020053, 2.06715429261956956760919668483, 2.14655873873253757769721274675, 2.26339410567486130003182245167, 2.26525933591875023635767300839, 2.67402538236873830691499900714, 2.69671423820908542755360543616, 2.79054051921709119995551940344, 2.95331654866925326455233834708, 2.97447454796092782650989884802, 2.99851932164108026702505154775, 3.28239734147959649140593305471, 3.38449370565176405325727462572, 3.41845638561531586576518534399, 3.46481160600584306661912146577, 3.57514782796243959163573400822, 3.58989539608701462091752793780, 3.69269407874133238887112002482

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

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