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Properties

Label 24-1044e12-1.1-c0e12-0-0
Degree $24$
Conductor $1.677\times 10^{36}$
Sign $1$
Analytic cond. $0.000400213$
Root an. cond. $0.721819$
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  + 4-s − 4·13-s + 2·25-s − 2·49-s − 4·52-s + 14·73-s − 14·97-s + 2·100-s − 10·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + ⋯
L(s)  = 1  + 4-s − 4·13-s + 2·25-s − 2·49-s − 4·52-s + 14·73-s − 14·97-s + 2·100-s − 10·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{24} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(0.000400213\)
Root analytic conductor: \(0.721819\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{24} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4815391247\)
\(L(\frac12)\) \(\approx\) \(0.4815391247\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
3 \( 1 \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
good5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
53 \( ( 1 + T^{2} )^{6}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
59 \( ( 1 - T )^{12}( 1 + T )^{12} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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 \( ( 1 - T )^{12}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
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 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
97 \( ( 1 + T )^{12}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
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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

−3.44910634637365981737457304547, −3.31927192945386137165189089143, −3.25664156111365629198770287496, −3.08157601856144236119350673143, −3.06364718604947944603517630796, −2.75466688350496495892879662539, −2.70231240094035605598239285402, −2.69423557331932131948451002854, −2.62420663887542020562666385464, −2.59816251088589322282275009436, −2.45841492275173339965052598435, −2.40904022631874351370430049965, −2.33742522378031155051489215453, −2.27103505064598543055921434528, −2.09321117338280748840886315387, −2.00282529688624234515291158718, −1.77051201290097105919044043809, −1.66822713734094875923289264307, −1.55965360603668715718980070702, −1.42697028967573913078945143534, −1.35904443797505262076684347848, −1.10568362167479118874559339972, −1.07842876368033985886516891147, −0.76788066343210876526571953791, −0.46853900830064246834725029555, 0.46853900830064246834725029555, 0.76788066343210876526571953791, 1.07842876368033985886516891147, 1.10568362167479118874559339972, 1.35904443797505262076684347848, 1.42697028967573913078945143534, 1.55965360603668715718980070702, 1.66822713734094875923289264307, 1.77051201290097105919044043809, 2.00282529688624234515291158718, 2.09321117338280748840886315387, 2.27103505064598543055921434528, 2.33742522378031155051489215453, 2.40904022631874351370430049965, 2.45841492275173339965052598435, 2.59816251088589322282275009436, 2.62420663887542020562666385464, 2.69423557331932131948451002854, 2.70231240094035605598239285402, 2.75466688350496495892879662539, 3.06364718604947944603517630796, 3.08157601856144236119350673143, 3.25664156111365629198770287496, 3.31927192945386137165189089143, 3.44910634637365981737457304547

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

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