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Properties

Label 24-3248e12-1.1-c0e12-0-6
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 2·7-s + 2·9-s + 10·11-s + 2·28-s + 2·36-s − 4·37-s + 10·44-s + 49-s − 2·53-s + 4·63-s − 2·67-s + 20·77-s − 12·79-s + 81-s + 20·99-s + 2·107-s − 2·109-s − 2·113-s + 55·121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 4-s + 2·7-s + 2·9-s + 10·11-s + 2·28-s + 2·36-s − 4·37-s + 10·44-s + 49-s − 2·53-s + 4·63-s − 2·67-s + 20·77-s − 12·79-s + 81-s + 20·99-s + 2·107-s − 2·109-s − 2·113-s + 55·121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-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

\(L(\frac{1}{2})\) \(\approx\) \(17.45834996\)
\(L(\frac12)\) \(\approx\) \(17.45834996\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
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 \( ( 1 - T )^{12}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \)
41 \( ( 1 + T^{4} )^{6} \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
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

−2.79751149489010250808362642237, −2.77688486667097467182393828053, −2.67425176524065091060864601262, −2.63397340456040291658417708635, −2.59914710941274936511872461965, −2.45312071749943158130974069066, −2.08988690239390530786101837173, −2.07881736572125152607530049236, −1.96540646633902955138197627146, −1.92217633095431054779979840584, −1.85030359561889331683260610976, −1.75721486935311685493786840912, −1.71738551139930188504464528703, −1.67013315026869209862449704218, −1.54520592885330843561989647062, −1.47858767408010069373361774764, −1.47543919360670917863784070353, −1.43778512286328160856667272549, −1.31247644239640193292139983634, −1.22239885763910415736143874009, −1.13668717223042360351857272186, −1.01697759784961173414890397869, −0.898294369263597626876827604463, −0.808116011630168211155561378650, −0.39269000606397556428996398437, 0.39269000606397556428996398437, 0.808116011630168211155561378650, 0.898294369263597626876827604463, 1.01697759784961173414890397869, 1.13668717223042360351857272186, 1.22239885763910415736143874009, 1.31247644239640193292139983634, 1.43778512286328160856667272549, 1.47543919360670917863784070353, 1.47858767408010069373361774764, 1.54520592885330843561989647062, 1.67013315026869209862449704218, 1.71738551139930188504464528703, 1.75721486935311685493786840912, 1.85030359561889331683260610976, 1.92217633095431054779979840584, 1.96540646633902955138197627146, 2.07881736572125152607530049236, 2.08988690239390530786101837173, 2.45312071749943158130974069066, 2.59914710941274936511872461965, 2.63397340456040291658417708635, 2.67425176524065091060864601262, 2.77688486667097467182393828053, 2.79751149489010250808362642237

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

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