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Properties

Label 24-63e24-1.1-c0e12-0-0
Degree $24$
Conductor $1.528\times 10^{43}$
Sign $1$
Analytic cond. $3648.03$
Root an. cond. $1.40740$
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 − 7-s + 3·13-s + 16-s − 2·25-s + 28-s − 3·31-s − 13·37-s + 43-s + 49-s − 3·52-s − 11·61-s − 67-s − 79-s − 3·91-s + 3·97-s + 2·100-s + 109-s − 112-s + 2·121-s + 3·124-s + 127-s + 131-s + 137-s + 139-s + 13·148-s + 149-s + ⋯
L(s)  = 1  − 4-s − 7-s + 3·13-s + 16-s − 2·25-s + 28-s − 3·31-s − 13·37-s + 43-s + 49-s − 3·52-s − 11·61-s − 67-s − 79-s − 3·91-s + 3·97-s + 2·100-s + 109-s − 112-s + 2·121-s + 3·124-s + 127-s + 131-s + 137-s + 139-s + 13·148-s + 149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{48} \cdot 7^{24}\)
Sign: $1$
Analytic conductor: \(3648.03\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{48} \cdot 7^{24} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1487418613\)
\(L(\frac12)\) \(\approx\) \(0.1487418613\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
good2 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
5 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
11 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
17 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
23 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
29 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
37 \( ( 1 + T )^{12}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
41 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
47 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
53 \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
59 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
61 \( ( 1 + T )^{12}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
83 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
89 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} ) \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} ) \)
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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

−2.87684814488145382697501710325, −2.86704678295177389637665995430, −2.73415781699611075637741205391, −2.34967935571848838356399775726, −2.30767757282308522270682861132, −2.26902817144006696465041027767, −2.20162142737553422307638610213, −2.19848002221238223474653054591, −2.08727851187436172123223841588, −1.86287207736807130923996757657, −1.83805941881446223057329949175, −1.80804860237661146562377715402, −1.73367153749849537062125931945, −1.61488707150331169677926293600, −1.47600724528512333738501594933, −1.43651905514544621111065399326, −1.43273452004547430724179965067, −1.36411242775700653958271615671, −1.25744009273168013388596891797, −1.25692908219981936707296279245, −1.01744317676668136906087102444, −0.49910378846716486912532331841, −0.48548408706461466005693011345, −0.40642214940556207912272534927, −0.14215645742292930269964619854, 0.14215645742292930269964619854, 0.40642214940556207912272534927, 0.48548408706461466005693011345, 0.49910378846716486912532331841, 1.01744317676668136906087102444, 1.25692908219981936707296279245, 1.25744009273168013388596891797, 1.36411242775700653958271615671, 1.43273452004547430724179965067, 1.43651905514544621111065399326, 1.47600724528512333738501594933, 1.61488707150331169677926293600, 1.73367153749849537062125931945, 1.80804860237661146562377715402, 1.83805941881446223057329949175, 1.86287207736807130923996757657, 2.08727851187436172123223841588, 2.19848002221238223474653054591, 2.20162142737553422307638610213, 2.26902817144006696465041027767, 2.30767757282308522270682861132, 2.34967935571848838356399775726, 2.73415781699611075637741205391, 2.86704678295177389637665995430, 2.87684814488145382697501710325

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

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