Properties

Label 24-72e24-1.1-c1e12-0-1
Degree $24$
Conductor $3.767\times 10^{44}$
Sign $1$
Analytic cond. $2.53107\times 10^{19}$
Root an. cond. $6.43385$
Motivic weight $1$
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  + 42·25-s + 36·41-s − 18·49-s + 48·89-s + 12·97-s + 60·113-s + 54·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 42·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 42/5·25-s + 5.62·41-s − 2.57·49-s + 5.08·89-s + 1.21·97-s + 5.64·113-s + 4.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{48}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{72} \cdot 3^{48}\)
Sign: $1$
Analytic conductor: \(2.53107\times 10^{19}\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5184} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{72} \cdot 3^{48} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(276.5915936\)
\(L(\frac12)\) \(\approx\) \(276.5915936\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{6} \)
7 \( ( 1 + 9 T^{2} + 90 T^{4} + 821 T^{6} + 90 p^{2} T^{8} + 9 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
11 \( ( 1 - 27 T^{2} + 270 T^{4} - 1987 T^{6} + 270 p^{2} T^{8} - 27 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
13 \( ( 1 - 21 T^{2} + 246 T^{4} - 3449 T^{6} + 246 p^{2} T^{8} - 21 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( ( 1 + 27 T^{2} - 36 T^{3} + 27 p T^{4} + p^{3} T^{6} )^{4} \)
19 \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{6} \)
23 \( ( 1 + 21 T^{2} + 66 p T^{4} + 22021 T^{6} + 66 p^{3} T^{8} + 21 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - 21 T^{2} + 2022 T^{4} - 36745 T^{6} + 2022 p^{2} T^{8} - 21 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 + 69 T^{2} + 2526 T^{4} + 73829 T^{6} + 2526 p^{2} T^{8} + 69 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 162 T^{2} + 12663 T^{4} - 590924 T^{6} + 12663 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
41 \( ( 1 - 9 T + 126 T^{2} - 729 T^{3} + 126 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
43 \( ( 1 - 135 T^{2} + 198 p T^{4} - 385099 T^{6} + 198 p^{3} T^{8} - 135 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( ( 1 - 15 T^{2} + 4434 T^{4} - 26651 T^{6} + 4434 p^{2} T^{8} - 15 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 138 T^{2} + 5703 T^{4} - 126556 T^{6} + 5703 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 - 207 T^{2} + 18234 T^{4} - 1120507 T^{6} + 18234 p^{2} T^{8} - 207 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 213 T^{2} + 21750 T^{4} - 1517753 T^{6} + 21750 p^{2} T^{8} - 213 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 171 T^{2} + 13518 T^{4} - 858931 T^{6} + 13518 p^{2} T^{8} - 171 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
71 \( ( 1 + 138 T^{2} + 13695 T^{4} + 1144348 T^{6} + 13695 p^{2} T^{8} + 138 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 + 135 T^{2} - 164 T^{3} + 135 p T^{4} + p^{3} T^{6} )^{4} \)
79 \( ( 1 + 381 T^{2} + 65118 T^{4} + 6516893 T^{6} + 65118 p^{2} T^{8} + 381 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
83 \( ( 1 - 327 T^{2} + 49722 T^{4} - 4885459 T^{6} + 49722 p^{2} T^{8} - 327 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 12 T + 231 T^{2} - 2028 T^{3} + 231 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
97 \( ( 1 - 3 T + 78 T^{2} + 605 T^{3} + 78 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
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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

−2.46622424288698084455936897982, −2.33095571310737458410114255027, −2.14810490877487650784198203056, −2.14778648746021920099650509885, −2.07323284975587755308245449176, −2.00804834558112798938264681832, −1.99950505746416678884163359544, −1.91019295508183477551283082227, −1.86071904570206749859106116509, −1.79892129219009776775352599165, −1.53724134964069856032267174280, −1.47826709801146549463717536680, −1.42758843337733831614443147408, −1.29330298167760678930966722713, −1.08636370589864071113434304085, −1.02437140318962088342965296932, −0.994528135201545335408882050102, −0.909925093846414396776528693836, −0.71215079956613412659487756453, −0.71039204617802575300516597231, −0.69705600768052912458525341211, −0.54025154102522291117917850156, −0.52878247858443635999136782237, −0.38555569587148034280660687511, −0.32644302790460736632170400537, 0.32644302790460736632170400537, 0.38555569587148034280660687511, 0.52878247858443635999136782237, 0.54025154102522291117917850156, 0.69705600768052912458525341211, 0.71039204617802575300516597231, 0.71215079956613412659487756453, 0.909925093846414396776528693836, 0.994528135201545335408882050102, 1.02437140318962088342965296932, 1.08636370589864071113434304085, 1.29330298167760678930966722713, 1.42758843337733831614443147408, 1.47826709801146549463717536680, 1.53724134964069856032267174280, 1.79892129219009776775352599165, 1.86071904570206749859106116509, 1.91019295508183477551283082227, 1.99950505746416678884163359544, 2.00804834558112798938264681832, 2.07323284975587755308245449176, 2.14778648746021920099650509885, 2.14810490877487650784198203056, 2.33095571310737458410114255027, 2.46622424288698084455936897982

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

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