Properties

Label 24-741e12-1.1-c0e12-0-1
Degree $24$
Conductor $2.740\times 10^{34}$
Sign $1$
Analytic cond. $6.54190\times 10^{-6}$
Root an. cond. $0.608117$
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  − 6·19-s + 6·43-s + 6·67-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  − 6·19-s + 6·43-s + 6·67-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 13^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 13^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 13^{12} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(6.54190\times 10^{-6}\)
Root analytic conductor: \(0.608117\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{741} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 13^{12} \cdot 19^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2884051461\)
\(L(\frac12)\) \(\approx\) \(0.2884051461\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{6} + T^{12} \)
13 \( ( 1 + T^{2} )^{6} \)
19 \( ( 1 + T + T^{2} )^{6} \)
good2 \( 1 - T^{12} + T^{24} \)
5 \( 1 - T^{12} + T^{24} \)
7 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
11 \( ( 1 - T^{4} + T^{8} )^{3} \)
17 \( ( 1 - T^{6} + T^{12} )^{2} \)
23 \( ( 1 - T^{6} + T^{12} )^{2} \)
29 \( ( 1 - T^{6} + T^{12} )^{2} \)
31 \( ( 1 - T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
37 \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
41 \( 1 - T^{12} + T^{24} \)
43 \( ( 1 - T + T^{2} )^{6}( 1 + T^{3} + T^{6} )^{2} \)
47 \( 1 - T^{12} + T^{24} \)
53 \( ( 1 - T^{6} + T^{12} )^{2} \)
59 \( 1 - T^{12} + T^{24} \)
61 \( ( 1 + T^{2} )^{6}( 1 - T^{6} + T^{12} ) \)
67 \( ( 1 - T + T^{2} )^{6}( 1 - T^{6} + T^{12} ) \)
71 \( 1 - T^{12} + T^{24} \)
73 \( ( 1 + T^{2} )^{6}( 1 - T^{3} + T^{6} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} )^{3}( 1 - T^{6} + T^{12} ) \)
83 \( ( 1 - T^{4} + T^{8} )^{3} \)
89 \( 1 - T^{12} + T^{24} \)
97 \( ( 1 - T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
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

−3.68614961978149697370999633315, −3.60961627479438716318016888269, −3.52808219707590905332663002961, −3.35226447619447485386909286200, −3.22364034227347435416600095936, −3.21494292381304427775711432027, −3.10645235030402900905115029794, −2.95872995667629680962436626628, −2.62899095947104205569330002627, −2.52742052660143094002292798868, −2.50041437650022521913206227136, −2.43456634565438528968226942411, −2.42930304717084765692415629598, −2.41612738441632790503036071495, −2.18725875094617455590997954860, −2.15068706508724126479813625664, −2.03039262886764307944222991015, −1.96477028634056906880683122112, −1.87833491200017237769111075480, −1.43489654544052631770358055294, −1.27890985919305542993301940965, −1.24750385238677628236073828661, −1.22001546226677557648525299933, −0.821051901606721378017881589489, −0.68923227797130601405362714830, 0.68923227797130601405362714830, 0.821051901606721378017881589489, 1.22001546226677557648525299933, 1.24750385238677628236073828661, 1.27890985919305542993301940965, 1.43489654544052631770358055294, 1.87833491200017237769111075480, 1.96477028634056906880683122112, 2.03039262886764307944222991015, 2.15068706508724126479813625664, 2.18725875094617455590997954860, 2.41612738441632790503036071495, 2.42930304717084765692415629598, 2.43456634565438528968226942411, 2.50041437650022521913206227136, 2.52742052660143094002292798868, 2.62899095947104205569330002627, 2.95872995667629680962436626628, 3.10645235030402900905115029794, 3.21494292381304427775711432027, 3.22364034227347435416600095936, 3.35226447619447485386909286200, 3.52808219707590905332663002961, 3.60961627479438716318016888269, 3.68614961978149697370999633315

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

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