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Properties

Label 24-2268e12-1.1-c1e12-0-1
Degree $24$
Conductor $1.852\times 10^{40}$
Sign $1$
Analytic cond. $1.24463\times 10^{15}$
Root an. cond. $4.25559$
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  − 18·19-s + 15·25-s − 24·37-s + 6·43-s + 54·73-s − 48·79-s + 36·97-s − 18·103-s − 12·109-s − 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 4.12·19-s + 3·25-s − 3.94·37-s + 0.914·43-s + 6.32·73-s − 5.40·79-s + 3.65·97-s − 1.77·103-s − 1.14·109-s − 1.90·121-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 − 2.76·169-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 + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(2^{24} \cdot 3^{48} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.24463\times 10^{15}\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{24} \cdot 3^{48} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.028531140\)
\(L(\frac12)\) \(\approx\) \(2.028531140\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( ( 1 - p T^{3} + p^{3} T^{6} )^{2} \)
good5 \( 1 - 3 p T^{2} + 96 T^{4} - 409 T^{6} + 1971 T^{8} - 11052 T^{10} + 56769 T^{12} - 11052 p^{2} T^{14} + 1971 p^{4} T^{16} - 409 p^{6} T^{18} + 96 p^{8} T^{20} - 3 p^{11} T^{22} + p^{12} T^{24} \)
11 \( 1 + 21 T^{2} + 120 T^{4} - 1085 T^{6} - 17973 T^{8} - 67788 T^{10} - 69951 T^{12} - 67788 p^{2} T^{14} - 17973 p^{4} T^{16} - 1085 p^{6} T^{18} + 120 p^{8} T^{20} + 21 p^{10} T^{22} + p^{12} T^{24} \)
13 \( ( 1 + 18 T^{2} + 90 T^{4} - 1134 T^{5} + 1141 T^{6} - 1134 p T^{7} + 90 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 + 12 T^{2} - 624 T^{4} - 4882 T^{6} + 283284 T^{8} + 1089036 T^{10} - 84522621 T^{12} + 1089036 p^{2} T^{14} + 283284 p^{4} T^{16} - 4882 p^{6} T^{18} - 624 p^{8} T^{20} + 12 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 + 9 T + 72 T^{2} + 405 T^{3} + 117 p T^{4} + 12546 T^{5} + 56077 T^{6} + 12546 p T^{7} + 117 p^{3} T^{8} + 405 p^{3} T^{9} + 72 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 84 T^{2} + 3684 T^{4} + 103558 T^{6} + 2039616 T^{8} + 26787600 T^{10} + 378704571 T^{12} + 26787600 p^{2} T^{14} + 2039616 p^{4} T^{16} + 103558 p^{6} T^{18} + 3684 p^{8} T^{20} + 84 p^{10} T^{22} + p^{12} T^{24} \)
29 \( 1 + 21 T^{2} - 2040 T^{4} - 16205 T^{6} + 3351627 T^{8} + 11952612 T^{10} - 3132495471 T^{12} + 11952612 p^{2} T^{14} + 3351627 p^{4} T^{16} - 16205 p^{6} T^{18} - 2040 p^{8} T^{20} + 21 p^{10} T^{22} + p^{12} T^{24} \)
31 \( ( 1 - 60 T^{2} + 2760 T^{4} - 114059 T^{6} + 2760 p^{2} T^{8} - 60 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 + 12 T + 48 T^{2} + 130 T^{3} - 468 T^{4} - 19764 T^{5} - 175701 T^{6} - 19764 p T^{7} - 468 p^{2} T^{8} + 130 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( 1 - 60 T^{2} - 1236 T^{4} + 102974 T^{6} + 3463560 T^{8} - 144625752 T^{10} - 1347317805 T^{12} - 144625752 p^{2} T^{14} + 3463560 p^{4} T^{16} + 102974 p^{6} T^{18} - 1236 p^{8} T^{20} - 60 p^{10} T^{22} + p^{12} T^{24} \)
43 \( ( 1 - 3 T - 102 T^{2} + 157 T^{3} + 6813 T^{4} - 4104 T^{5} - 327405 T^{6} - 4104 p T^{7} + 6813 p^{2} T^{8} + 157 p^{3} T^{9} - 102 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( ( 1 + 267 T^{2} + 30369 T^{4} + 1882699 T^{6} + 30369 p^{2} T^{8} + 267 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( 1 + 84 T^{2} + 2136 T^{4} - 35042 T^{6} - 8443404 T^{8} - 556134516 T^{10} - 29094135357 T^{12} - 556134516 p^{2} T^{14} - 8443404 p^{4} T^{16} - 35042 p^{6} T^{18} + 2136 p^{8} T^{20} + 84 p^{10} T^{22} + p^{12} T^{24} \)
59 \( ( 1 + 51 T^{2} + 3981 T^{4} + 180691 T^{6} + 3981 p^{2} T^{8} + 51 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 189 T^{2} + 16686 T^{4} - 1060409 T^{6} + 16686 p^{2} T^{8} - 189 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
67 \( ( 1 + 54 T^{2} - 497 T^{3} + 54 p T^{4} + p^{3} T^{6} )^{4} \)
71 \( ( 1 - 237 T^{2} + 29877 T^{4} - 2533201 T^{6} + 29877 p^{2} T^{8} - 237 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 27 T + 522 T^{2} - 7533 T^{3} + 94005 T^{4} - 988578 T^{5} + 9101239 T^{6} - 988578 p T^{7} + 94005 p^{2} T^{8} - 7533 p^{3} T^{9} + 522 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 + 12 T + 222 T^{2} + 1519 T^{3} + 222 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( 1 - 123 T^{2} + 2544 T^{4} + 684275 T^{6} - 65597985 T^{8} + 1444043232 T^{10} + 90501353529 T^{12} + 1444043232 p^{2} T^{14} - 65597985 p^{4} T^{16} + 684275 p^{6} T^{18} + 2544 p^{8} T^{20} - 123 p^{10} T^{22} + p^{12} T^{24} \)
89 \( ( 1 - 151 T^{2} + 14880 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} )^{3} \)
97 \( ( 1 - 18 T + 351 T^{2} - 4374 T^{3} + 56862 T^{4} - 696690 T^{5} + 7120267 T^{6} - 696690 p T^{7} + 56862 p^{2} T^{8} - 4374 p^{3} T^{9} + 351 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
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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

−2.75428440519052260880745795568, −2.71314973008676497809171197879, −2.64580599870196887302458268057, −2.36783667042032069011612955669, −2.36396820112982083387791038306, −2.33223757003595233080658031485, −2.19977714627142705693705814871, −2.12494478986355584965820138276, −2.09895984866289718289004328597, −1.95754657210728135866143510397, −1.92437097678773418725631770168, −1.86678963557726490159718749865, −1.71188387910907697019781809564, −1.60369437219103415237251246216, −1.28008974447702746447955656747, −1.23641678797394765547207290693, −1.22898858763804716119180774768, −1.12363661231682330653699654504, −1.05669949422575879739781534770, −1.03550512707653397365411287925, −0.73614468930319055385945416149, −0.44673875459435172138756392839, −0.31550303002076529897974302364, −0.21078010371055272768071614409, −0.17298619326801126627999374407, 0.17298619326801126627999374407, 0.21078010371055272768071614409, 0.31550303002076529897974302364, 0.44673875459435172138756392839, 0.73614468930319055385945416149, 1.03550512707653397365411287925, 1.05669949422575879739781534770, 1.12363661231682330653699654504, 1.22898858763804716119180774768, 1.23641678797394765547207290693, 1.28008974447702746447955656747, 1.60369437219103415237251246216, 1.71188387910907697019781809564, 1.86678963557726490159718749865, 1.92437097678773418725631770168, 1.95754657210728135866143510397, 2.09895984866289718289004328597, 2.12494478986355584965820138276, 2.19977714627142705693705814871, 2.33223757003595233080658031485, 2.36396820112982083387791038306, 2.36783667042032069011612955669, 2.64580599870196887302458268057, 2.71314973008676497809171197879, 2.75428440519052260880745795568

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

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