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Properties

Label 24-38e24-1.1-c0e12-0-1
Degree $24$
Conductor $8.219\times 10^{37}$
Sign $1$
Analytic cond. $0.0196196$
Root an. cond. $0.848910$
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  − 6·7-s + 6·11-s + 21·49-s − 36·77-s + 21·121-s + 4·125-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 + 233-s + ⋯
L(s)  = 1  − 6·7-s + 6·11-s + 21·49-s − 36·77-s + 21·121-s + 4·125-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 + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 19^{24}\)
Sign: $1$
Analytic conductor: \(0.0196196\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 19^{24} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7586673312\)
\(L(\frac12)\) \(\approx\) \(0.7586673312\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−3.26570676132359359108466865379, −3.22985730346293703550190729864, −3.13352908739783390120440584303, −3.13323400073370687336146945923, −3.05602487638446396071454604385, −2.87335052574645884269982823555, −2.70470684714235883219210567213, −2.62287178525519226845103481225, −2.47239845984761214581065418652, −2.35637963161640333326001896865, −2.35591308637796925960359329067, −2.26110267207529236848524944744, −2.23528097981416535904261188038, −2.05429710546338401578347421757, −1.85495336278615597046951111113, −1.70804044143355649779701917753, −1.59500835757038432286176786786, −1.56373271096809404512043635080, −1.40723721851488388117922223271, −1.22116492764177397244299601523, −1.18872135141460663158234252029, −0.855263366079335504856861325321, −0.817372157785485705685165283926, −0.69686514898170891438626966884, −0.60234090160508946529768394078, 0.60234090160508946529768394078, 0.69686514898170891438626966884, 0.817372157785485705685165283926, 0.855263366079335504856861325321, 1.18872135141460663158234252029, 1.22116492764177397244299601523, 1.40723721851488388117922223271, 1.56373271096809404512043635080, 1.59500835757038432286176786786, 1.70804044143355649779701917753, 1.85495336278615597046951111113, 2.05429710546338401578347421757, 2.23528097981416535904261188038, 2.26110267207529236848524944744, 2.35591308637796925960359329067, 2.35637963161640333326001896865, 2.47239845984761214581065418652, 2.62287178525519226845103481225, 2.70470684714235883219210567213, 2.87335052574645884269982823555, 3.05602487638446396071454604385, 3.13323400073370687336146945923, 3.13352908739783390120440584303, 3.22985730346293703550190729864, 3.26570676132359359108466865379

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

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