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Properties

Label 24-2340e12-1.1-c1e12-0-2
Degree $24$
Conductor $2.695\times 10^{40}$
Sign $1$
Analytic cond. $1.81098\times 10^{15}$
Root an. cond. $4.32261$
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  − 8·7-s + 12·13-s − 24·31-s − 12·37-s + 32·49-s − 32·61-s + 40·67-s − 12·73-s + 8·79-s − 96·91-s − 28·97-s + 12·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 3.02·7-s + 3.32·13-s − 4.31·31-s − 1.97·37-s + 32/7·49-s − 4.09·61-s + 4.88·67-s − 1.40·73-s + 0.900·79-s − 10.0·91-s − 2.84·97-s + 1.14·109-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 + 2.61·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(1.81098\times 10^{15}\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5427006759\)
\(L(\frac12)\) \(\approx\) \(0.5427006759\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( ( 1 + T^{4} )^{3} \)
13 \( ( 1 - 6 T + 37 T^{2} - 120 T^{3} + 37 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
good7 \( ( 1 + 4 T + 8 T^{2} + 24 T^{3} - 48 T^{4} - 356 T^{5} - 752 T^{6} - 356 p T^{7} - 48 p^{2} T^{8} + 24 p^{3} T^{9} + 8 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
11 \( 1 + 224 T^{4} + 34112 T^{8} + 4281170 T^{12} + 34112 p^{4} T^{16} + 224 p^{8} T^{20} + p^{12} T^{24} \)
17 \( ( 1 + 54 T^{2} + 1756 T^{4} + 35276 T^{6} + 1756 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 + p^{2} T^{4} )^{6} \)
23 \( ( 1 + 100 T^{2} + 4776 T^{4} + 138222 T^{6} + 4776 p^{2} T^{8} + 100 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - 148 T^{2} + 9759 T^{4} - 365832 T^{6} + 9759 p^{2} T^{8} - 148 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 + 12 T + 72 T^{2} + 468 T^{3} + 2403 T^{4} + 8760 T^{5} + 41616 T^{6} + 8760 p T^{7} + 2403 p^{2} T^{8} + 468 p^{3} T^{9} + 72 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( ( 1 + 6 T + 18 T^{2} + 218 T^{3} - 1080 T^{4} - 14478 T^{5} - 43666 T^{6} - 14478 p T^{7} - 1080 p^{2} T^{8} + 218 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( 1 - 2976 T^{4} - 2260928 T^{8} + 14921687522 T^{12} - 2260928 p^{4} T^{16} - 2976 p^{8} T^{20} + p^{12} T^{24} \)
43 \( ( 1 - 122 T^{2} + 7431 T^{4} - 346668 T^{6} + 7431 p^{2} T^{8} - 122 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( 1 + 3582 T^{4} + 519615 T^{8} - 13446532348 T^{12} + 519615 p^{4} T^{16} + 3582 p^{8} T^{20} + p^{12} T^{24} \)
53 \( ( 1 + 50 T^{2} + 8916 T^{4} + 277452 T^{6} + 8916 p^{2} T^{8} + 50 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( 1 - 2794 T^{4} + 8970143 T^{8} + 12680524724 T^{12} + 8970143 p^{4} T^{16} - 2794 p^{8} T^{20} + p^{12} T^{24} \)
61 \( ( 1 + 8 T + 60 T^{2} + 414 T^{3} + 60 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
67 \( ( 1 - 20 T + 200 T^{2} - 1916 T^{3} + 14555 T^{4} - 83240 T^{5} + 589328 T^{6} - 83240 p T^{7} + 14555 p^{2} T^{8} - 1916 p^{3} T^{9} + 200 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( 1 + 2064 T^{4} - 1496208 T^{8} + 21009924322 T^{12} - 1496208 p^{4} T^{16} + 2064 p^{8} T^{20} + p^{12} T^{24} \)
73 \( ( 1 + 6 T + 18 T^{2} - 1290 T^{3} - 5133 T^{4} + 39180 T^{5} + 1159524 T^{6} + 39180 p T^{7} - 5133 p^{2} T^{8} - 1290 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 2 T + 118 T^{2} - 744 T^{3} + 118 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( 1 + 33870 T^{4} + 522018799 T^{8} + 4622798619044 T^{12} + 522018799 p^{4} T^{16} + 33870 p^{8} T^{20} + p^{12} T^{24} \)
89 \( 1 + 13696 T^{4} + 9220256 T^{8} - 673229407262 T^{12} + 9220256 p^{4} T^{16} + 13696 p^{8} T^{20} + p^{12} T^{24} \)
97 \( ( 1 + 14 T + 98 T^{2} + 1042 T^{3} + 9360 T^{4} + 73258 T^{5} + 651214 T^{6} + 73258 p T^{7} + 9360 p^{2} T^{8} + 1042 p^{3} T^{9} + 98 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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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

−2.77161054544584852834528285909, −2.72741624660827136574963887968, −2.57234734762168803949056417025, −2.46431866426013555266441200084, −2.36432674546193336020127266181, −2.31355981957411568101802253437, −2.26635536907510033669641495810, −2.23349782102867190600727343072, −2.16751084184601290261793159367, −1.73694002006039958697294911457, −1.72286110434525696618118580907, −1.70387467559961413143476418293, −1.58366534634477908255822078751, −1.53027247604696081206216313398, −1.52227333398291535364130008285, −1.44757193180711689632499579572, −1.32910044963240092195617129233, −1.02939938527994315464428028118, −0.906153964738890413076055563108, −0.852732926387324783470819514247, −0.75333384362968107670387485333, −0.42891748469140301240690601163, −0.36800061843983229353483439457, −0.27686109984499898862121913589, −0.06898545807669371809415884370, 0.06898545807669371809415884370, 0.27686109984499898862121913589, 0.36800061843983229353483439457, 0.42891748469140301240690601163, 0.75333384362968107670387485333, 0.852732926387324783470819514247, 0.906153964738890413076055563108, 1.02939938527994315464428028118, 1.32910044963240092195617129233, 1.44757193180711689632499579572, 1.52227333398291535364130008285, 1.53027247604696081206216313398, 1.58366534634477908255822078751, 1.70387467559961413143476418293, 1.72286110434525696618118580907, 1.73694002006039958697294911457, 2.16751084184601290261793159367, 2.23349782102867190600727343072, 2.26635536907510033669641495810, 2.31355981957411568101802253437, 2.36432674546193336020127266181, 2.46431866426013555266441200084, 2.57234734762168803949056417025, 2.72741624660827136574963887968, 2.77161054544584852834528285909

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

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