Properties

Label 24-740e12-1.1-c0e12-0-0
Degree $24$
Conductor $2.696\times 10^{34}$
Sign $1$
Analytic cond. $6.43674\times 10^{-6}$
Root an. cond. $0.607707$
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  − 6·17-s + 3·25-s − 6·41-s − 12·61-s + 64-s + 6·73-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 6·17-s + 3·25-s − 6·41-s − 12·61-s + 64-s + 6·73-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 5^{12} \cdot 37^{12}\)
Sign: $1$
Analytic conductor: \(6.43674\times 10^{-6}\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 5^{12} \cdot 37^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07624344228\)
\(L(\frac12)\) \(\approx\) \(0.07624344228\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{6} + T^{12} \)
5 \( ( 1 - T^{2} + T^{4} )^{3} \)
37 \( 1 - T^{6} + T^{12} \)
good3 \( 1 - T^{12} + T^{24} \)
7 \( 1 - T^{12} + T^{24} \)
11 \( ( 1 - T^{2} + T^{4} )^{6} \)
13 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
17 \( ( 1 + T + T^{2} )^{6}( 1 + T^{3} + T^{6} )^{2} \)
19 \( 1 - T^{12} + T^{24} \)
23 \( ( 1 - T^{2} + T^{4} )^{6} \)
29 \( ( 1 - T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
31 \( ( 1 + T^{4} )^{6} \)
41 \( ( 1 + T + T^{2} )^{6}( 1 - T^{3} + T^{6} )^{2} \)
43 \( ( 1 + T^{2} )^{12} \)
47 \( ( 1 - T^{4} + T^{8} )^{3} \)
53 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
59 \( 1 - T^{12} + T^{24} \)
61 \( ( 1 + T )^{12}( 1 - T^{6} + T^{12} ) \)
67 \( 1 - T^{12} + T^{24} \)
71 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
73 \( ( 1 - T + T^{2} )^{6}( 1 - T^{2} + T^{4} )^{3} \)
79 \( 1 - T^{12} + T^{24} \)
83 \( 1 - T^{12} + T^{24} \)
89 \( ( 1 + T^{2} )^{6}( 1 + T^{3} + T^{6} )^{2} \)
97 \( ( 1 - T^{6} + T^{12} )^{2} \)
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

−3.56630428540586114876625315696, −3.41118207752876735468449731671, −3.38828621985639590162423941445, −3.36184895739625839583905019400, −3.31002131445263096759138357195, −3.27384209442318755550405282664, −3.16738692510039186285834946474, −2.94798617079922184621483765315, −2.93857423065377951281780404827, −2.58921980027080127982068932535, −2.52320938570822896644558510632, −2.46081872740519679359040361226, −2.35959192368561298468016594127, −2.30757495317234191123677241546, −2.20053531237598777928723014866, −2.19702367912220988542991785592, −1.95533852789938805872682765108, −1.89788314147824459590205165907, −1.62788211661275319159173968068, −1.57231095733345363035232617912, −1.36393930898467415438610800549, −1.34764532820558456527781263646, −1.15515603179824714247334384741, −1.08779499773764868223892473897, −0.26425341263058568824718342544, 0.26425341263058568824718342544, 1.08779499773764868223892473897, 1.15515603179824714247334384741, 1.34764532820558456527781263646, 1.36393930898467415438610800549, 1.57231095733345363035232617912, 1.62788211661275319159173968068, 1.89788314147824459590205165907, 1.95533852789938805872682765108, 2.19702367912220988542991785592, 2.20053531237598777928723014866, 2.30757495317234191123677241546, 2.35959192368561298468016594127, 2.46081872740519679359040361226, 2.52320938570822896644558510632, 2.58921980027080127982068932535, 2.93857423065377951281780404827, 2.94798617079922184621483765315, 3.16738692510039186285834946474, 3.27384209442318755550405282664, 3.31002131445263096759138357195, 3.36184895739625839583905019400, 3.38828621985639590162423941445, 3.41118207752876735468449731671, 3.56630428540586114876625315696

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

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