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Properties

Label 24-48e12-1.1-c1e12-0-0
Degree $24$
Conductor $1.496\times 10^{20}$
Sign $1$
Analytic cond. $1.00512\times 10^{-5}$
Root an. cond. $0.619097$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 8·7-s + 2·9-s + 4·12-s − 4·13-s + 6·16-s − 12·19-s + 16·21-s + 2·27-s + 16·28-s − 4·36-s − 4·37-s + 8·39-s + 12·43-s − 12·48-s − 20·49-s + 8·52-s + 24·57-s + 12·61-s − 16·63-s + 28·67-s + 24·76-s − 7·81-s − 32·84-s + 32·91-s − 8·97-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 3.02·7-s + 2/3·9-s + 1.15·12-s − 1.10·13-s + 3/2·16-s − 2.75·19-s + 3.49·21-s + 0.384·27-s + 3.02·28-s − 2/3·36-s − 0.657·37-s + 1.28·39-s + 1.82·43-s − 1.73·48-s − 2.85·49-s + 1.10·52-s + 3.17·57-s + 1.53·61-s − 2.01·63-s + 3.42·67-s + 2.75·76-s − 7/9·81-s − 3.49·84-s + 3.35·91-s − 0.812·97-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(1.00512\times 10^{-5}\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 3^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.04722424101\)
\(L(\frac12)\) \(\approx\) \(0.04722424101\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T^{2} - p T^{4} - p^{4} T^{6} - p^{3} T^{8} + p^{5} T^{10} + p^{6} T^{12} \)
3 \( 1 + 2 T + 2 T^{2} - 2 T^{3} - 5 T^{4} - 20 T^{5} - 28 T^{6} - 20 p T^{7} - 5 p^{2} T^{8} - 2 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
good5 \( 1 - 6 p T^{4} - 49 T^{8} + 12796 T^{12} - 49 p^{4} T^{16} - 6 p^{9} T^{20} + p^{12} T^{24} \)
7 \( ( 1 + 2 T + 15 T^{2} + 20 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
11 \( 1 - 62 T^{4} + 11023 T^{8} - 2631620 T^{12} + 11023 p^{4} T^{16} - 62 p^{8} T^{20} + p^{12} T^{24} \)
13 \( ( 1 + 2 T + 2 T^{2} - 6 T^{3} - 25 T^{4} + 412 T^{5} + 892 T^{6} + 412 p T^{7} - 25 p^{2} T^{8} - 6 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
17 \( ( 1 - 62 T^{2} + 1903 T^{4} - 38180 T^{6} + 1903 p^{2} T^{8} - 62 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 + 6 T + 18 T^{2} + 82 T^{3} + 539 T^{4} + 2636 T^{5} + 9476 T^{6} + 2636 p T^{7} + 539 p^{2} T^{8} + 82 p^{3} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( ( 1 - 86 T^{2} + 3791 T^{4} - 105684 T^{6} + 3791 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( 1 - 830 T^{4} + 2253679 T^{8} - 1165110596 T^{12} + 2253679 p^{4} T^{16} - 830 p^{8} T^{20} + p^{12} T^{24} \)
31 \( ( 1 - 150 T^{2} + 10019 T^{4} - 392444 T^{6} + 10019 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 + 2 T + 2 T^{2} - 54 T^{3} + 567 T^{4} + 8764 T^{5} + 17852 T^{6} + 8764 p T^{7} + 567 p^{2} T^{8} - 54 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( ( 1 + 138 T^{2} + 7887 T^{4} + 320492 T^{6} + 7887 p^{2} T^{8} + 138 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 - 6 T + 18 T^{2} - 226 T^{3} + 4235 T^{4} - 18188 T^{5} + 58436 T^{6} - 18188 p T^{7} + 4235 p^{2} T^{8} - 226 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( ( 1 + 170 T^{2} + 15791 T^{4} + 908172 T^{6} + 15791 p^{2} T^{8} + 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( 1 + 7714 T^{4} + 19237903 T^{8} + 30633057916 T^{12} + 19237903 p^{4} T^{16} + 7714 p^{8} T^{20} + p^{12} T^{24} \)
59 \( ( 1 - 30 T + 450 T^{2} - 3458 T^{3} + 3915 T^{4} + 231740 T^{5} - 2735068 T^{6} + 231740 p T^{7} + 3915 p^{2} T^{8} - 3458 p^{3} T^{9} + 450 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} )( 1 + 30 T + 450 T^{2} + 3458 T^{3} + 3915 T^{4} - 231740 T^{5} - 2735068 T^{6} - 231740 p T^{7} + 3915 p^{2} T^{8} + 3458 p^{3} T^{9} + 450 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} ) \)
61 \( ( 1 - 6 T + 18 T^{2} - 430 T^{3} - 121 T^{4} + 30796 T^{5} - 90148 T^{6} + 30796 p T^{7} - 121 p^{2} T^{8} - 430 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 14 T + 98 T^{2} - 706 T^{3} + 9435 T^{4} - 112164 T^{5} + 894884 T^{6} - 112164 p T^{7} + 9435 p^{2} T^{8} - 706 p^{3} T^{9} + 98 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 230 T^{2} + 30127 T^{4} - 2527028 T^{6} + 30127 p^{2} T^{8} - 230 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 166 T^{2} + 19007 T^{4} - 1414164 T^{6} + 19007 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 358 T^{2} + 58915 T^{4} - 5817628 T^{6} + 58915 p^{2} T^{8} - 358 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
83 \( 1 - 1374 T^{4} + 18563631 T^{8} - 336062521604 T^{12} + 18563631 p^{4} T^{16} - 1374 p^{8} T^{20} + p^{12} T^{24} \)
89 \( ( 1 + 322 T^{2} + 51919 T^{4} + 5548348 T^{6} + 51919 p^{2} T^{8} + 322 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( ( 1 + 2 T + 163 T^{2} - 220 T^{3} + 163 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
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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

−6.16292571127716224411723999073, −6.08857269367458711966180965223, −5.83526797377675345309218897203, −5.83189739480095664314390603882, −5.63088384160470586525400218939, −5.39287300666777092378821769418, −5.36513628414595212095012969214, −5.18907827279778118711461963409, −5.05116096829299101764919662120, −4.86923829016358286536353615561, −4.63252960383607768840319553217, −4.60154615981983080022896872675, −4.22408828721365359568982694189, −4.21289449256991863971045032813, −4.19823136021397385132735328492, −3.96912143353736020292877361809, −3.49868648684200611022182584110, −3.49043408298866264078051416716, −3.23986236753145177440224526434, −3.18625564779949222248919522397, −3.10481947319434072200526019304, −2.62921195346481332736716364162, −2.29677960166614722649074286943, −2.19759953500604809690517725123, −1.69578359853725675531896880474, 1.69578359853725675531896880474, 2.19759953500604809690517725123, 2.29677960166614722649074286943, 2.62921195346481332736716364162, 3.10481947319434072200526019304, 3.18625564779949222248919522397, 3.23986236753145177440224526434, 3.49043408298866264078051416716, 3.49868648684200611022182584110, 3.96912143353736020292877361809, 4.19823136021397385132735328492, 4.21289449256991863971045032813, 4.22408828721365359568982694189, 4.60154615981983080022896872675, 4.63252960383607768840319553217, 4.86923829016358286536353615561, 5.05116096829299101764919662120, 5.18907827279778118711461963409, 5.36513628414595212095012969214, 5.39287300666777092378821769418, 5.63088384160470586525400218939, 5.83189739480095664314390603882, 5.83526797377675345309218897203, 6.08857269367458711966180965223, 6.16292571127716224411723999073

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

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