Properties

Label 24-812e12-1.1-c0e12-0-0
Degree $24$
Conductor $8.216\times 10^{34}$
Sign $1$
Analytic cond. $1.96136\times 10^{-5}$
Root an. cond. $0.636585$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·11-s + 2·25-s + 2·37-s − 2·43-s − 2·44-s + 49-s − 4·53-s − 4·71-s + 2·79-s + 81-s + 2·100-s − 12·113-s − 5·121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 2·172-s + 173-s + ⋯
L(s)  = 1  + 4-s − 2·11-s + 2·25-s + 2·37-s − 2·43-s − 2·44-s + 49-s − 4·53-s − 4·71-s + 2·79-s + 81-s + 2·100-s − 12·113-s − 5·121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 2·172-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 7^{12} \cdot 29^{12}\)
Sign: $1$
Analytic conductor: \(1.96136\times 10^{-5}\)
Root analytic conductor: \(0.636585\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{812} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 7^{12} \cdot 29^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2999655232\)
\(L(\frac12)\) \(\approx\) \(0.2999655232\)
\(L(1)\) 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 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
29 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} \)
good3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
11 \( ( 1 + T^{2} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
17 \( ( 1 + T^{4} )^{6} \)
19 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
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 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
41 \( ( 1 + T^{4} )^{6} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} ) \)
47 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \)
59 \( ( 1 + T^{2} )^{12} \)
61 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \)
73 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
79 \( ( 1 + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
83 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
89 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
97 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.55149447880551848252234994473, −3.45180347036660480651785408179, −3.43529835333751640041820124410, −3.23710067452370850880012468105, −3.17270784056991851752757865699, −2.97417866118920656412230067474, −2.92230956949161539974098551060, −2.85494472879012645280137423080, −2.77553334681957402410167372646, −2.68983676181695654199739229302, −2.55279023846172454195735615271, −2.52569970385034418522700388981, −2.52555906889447320529397394882, −2.30154461535614249809555404457, −2.12261647148939605707690552978, −2.07800981935279369729422793143, −1.97554176496666189944929993495, −1.88089198822838948224252592569, −1.43393579109640723762180099238, −1.37438915414813452530741630643, −1.34793220755697594101712136589, −1.29284724918220903271191856637, −1.28193244624654993512997623438, −1.10607570866523779145999323227, −0.43915235347409690407444232196, 0.43915235347409690407444232196, 1.10607570866523779145999323227, 1.28193244624654993512997623438, 1.29284724918220903271191856637, 1.34793220755697594101712136589, 1.37438915414813452530741630643, 1.43393579109640723762180099238, 1.88089198822838948224252592569, 1.97554176496666189944929993495, 2.07800981935279369729422793143, 2.12261647148939605707690552978, 2.30154461535614249809555404457, 2.52555906889447320529397394882, 2.52569970385034418522700388981, 2.55279023846172454195735615271, 2.68983676181695654199739229302, 2.77553334681957402410167372646, 2.85494472879012645280137423080, 2.92230956949161539974098551060, 2.97417866118920656412230067474, 3.17270784056991851752757865699, 3.23710067452370850880012468105, 3.43529835333751640041820124410, 3.45180347036660480651785408179, 3.55149447880551848252234994473

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

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