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Properties

Label 24-363e12-1.1-c2e12-0-0
Degree $24$
Conductor $5.235\times 10^{30}$
Sign $1$
Analytic cond. $8.76799\times 10^{11}$
Root an. cond. $3.14500$
Motivic weight $2$
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  − 4·3-s + 2·4-s + 2·9-s − 8·12-s − 19·16-s + 106·25-s − 64·27-s − 8·31-s + 4·36-s − 244·37-s + 76·48-s − 244·49-s − 192·64-s − 128·67-s − 424·75-s + 377·81-s + 32·93-s − 156·97-s + 212·100-s − 8·103-s − 128·108-s + 976·111-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 4/3·3-s + 1/2·4-s + 2/9·9-s − 2/3·12-s − 1.18·16-s + 4.23·25-s − 2.37·27-s − 0.258·31-s + 1/9·36-s − 6.59·37-s + 1.58·48-s − 4.97·49-s − 3·64-s − 1.91·67-s − 5.65·75-s + 4.65·81-s + 0.344·93-s − 1.60·97-s + 2.11·100-s − 0.0776·103-s − 1.18·108-s + 8.79·111-s − 0.129·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 11^{24}\)
Sign: $1$
Analytic conductor: \(8.76799\times 10^{11}\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 11^{24} ,\ ( \ : [1]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.03705657458\)
\(L(\frac12)\) \(\approx\) \(0.03705657458\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + 2 T + 5 T^{2} + 46 T^{3} + 5 p^{2} T^{4} + 2 p^{4} T^{5} + p^{6} T^{6} )^{2} \)
11 \( 1 \)
good2 \( ( 1 - T^{2} + 11 T^{4} + 65 T^{6} + 11 p^{4} T^{8} - p^{8} T^{10} + p^{12} T^{12} )^{2} \)
5 \( ( 1 - 53 T^{2} + 66 p^{2} T^{4} - 46753 T^{6} + 66 p^{6} T^{8} - 53 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
7 \( ( 1 + 122 T^{2} + 4947 T^{4} + 126760 T^{6} + 4947 p^{4} T^{8} + 122 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
13 \( ( 1 + 573 T^{2} + 180330 T^{4} + 37225577 T^{6} + 180330 p^{4} T^{8} + 573 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
17 \( ( 1 - 763 T^{2} + 358694 T^{4} - 113685055 T^{6} + 358694 p^{4} T^{8} - 763 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
19 \( ( 1 + 1250 T^{2} + 883587 T^{4} + 385488376 T^{6} + 883587 p^{4} T^{8} + 1250 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
23 \( ( 1 - 1790 T^{2} + 1798083 T^{4} - 1152865504 T^{6} + 1798083 p^{4} T^{8} - 1790 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
29 \( ( 1 - 4471 T^{2} + 8761790 T^{4} - 9598580095 T^{6} + 8761790 p^{4} T^{8} - 4471 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
31 \( ( 1 + 2 T + 2451 T^{2} + 1726 T^{3} + 2451 p^{2} T^{4} + 2 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
37 \( ( 1 + 61 T + 5138 T^{2} + 171103 T^{3} + 5138 p^{2} T^{4} + 61 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
41 \( ( 1 - 7507 T^{2} + 27188246 T^{4} - 57902035255 T^{6} + 27188246 p^{4} T^{8} - 7507 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
43 \( ( 1 + 2846 T^{2} + 11634351 T^{4} + 19160307604 T^{6} + 11634351 p^{4} T^{8} + 2846 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
47 \( ( 1 - 7526 T^{2} + 31420695 T^{4} - 84347336248 T^{6} + 31420695 p^{4} T^{8} - 7526 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
53 \( ( 1 - 9789 T^{2} + 52368306 T^{4} - 180655410665 T^{6} + 52368306 p^{4} T^{8} - 9789 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
59 \( ( 1 - 14010 T^{2} + 97931823 T^{4} - 421188333788 T^{6} + 97931823 p^{4} T^{8} - 14010 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
61 \( ( 1 + 15890 T^{2} + 122244831 T^{4} + 569517934396 T^{6} + 122244831 p^{4} T^{8} + 15890 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
67 \( ( 1 + 32 T + 6375 T^{2} + 100726 T^{3} + 6375 p^{2} T^{4} + 32 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
71 \( ( 1 + 7858 T^{2} + 95414031 T^{4} + 413244154220 T^{6} + 95414031 p^{4} T^{8} + 7858 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
73 \( ( 1 + 12890 T^{2} + 108877647 T^{4} + 699115670524 T^{6} + 108877647 p^{4} T^{8} + 12890 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
79 \( ( 1 + 27954 T^{2} + 374627655 T^{4} + 2963157273056 T^{6} + 374627655 p^{4} T^{8} + 27954 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
83 \( ( 1 - 24262 T^{2} + 330028931 T^{4} - 33370259144 p T^{6} + 330028931 p^{4} T^{8} - 24262 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
89 \( ( 1 - 6725 T^{2} + 132991638 T^{4} - 763357844593 T^{6} + 132991638 p^{4} T^{8} - 6725 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
97 \( ( 1 + 39 T + 23274 T^{2} + 648667 T^{3} + 23274 p^{2} T^{4} + 39 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
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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

−3.64416542957247269723217275047, −3.61753028926905047929556901773, −3.38200892655922521586790846395, −3.12870742477057612013613783623, −3.12473720319859841876034743006, −3.03484141291380548028944765507, −2.90023968614848375897142594721, −2.79669619923347111092593392015, −2.75921139075967615633774935140, −2.71955752292475405397280248242, −2.39678250071812401006176192362, −2.26927366034122091926115734906, −1.94480993419065661402952168846, −1.91782878334438520544737697104, −1.81441267706322618610868174601, −1.67130794751689602561877756480, −1.66195332956059534308134708257, −1.60150975856027996411476577044, −1.36899729105193982361131991211, −1.09317522984858839973945894734, −0.922585019642760219447578344224, −0.76662869535222611198187164275, −0.28624078383815830014674128891, −0.18169977073853167936472318039, −0.05563948117819463881345999035, 0.05563948117819463881345999035, 0.18169977073853167936472318039, 0.28624078383815830014674128891, 0.76662869535222611198187164275, 0.922585019642760219447578344224, 1.09317522984858839973945894734, 1.36899729105193982361131991211, 1.60150975856027996411476577044, 1.66195332956059534308134708257, 1.67130794751689602561877756480, 1.81441267706322618610868174601, 1.91782878334438520544737697104, 1.94480993419065661402952168846, 2.26927366034122091926115734906, 2.39678250071812401006176192362, 2.71955752292475405397280248242, 2.75921139075967615633774935140, 2.79669619923347111092593392015, 2.90023968614848375897142594721, 3.03484141291380548028944765507, 3.12473720319859841876034743006, 3.12870742477057612013613783623, 3.38200892655922521586790846395, 3.61753028926905047929556901773, 3.64416542957247269723217275047

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

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