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Properties

Label 24-1323e12-1.1-c1e12-0-4
Degree $24$
Conductor $2.876\times 10^{37}$
Sign $1$
Analytic cond. $1.93216\times 10^{12}$
Root an. cond. $3.25026$
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  + 2·2-s + 5·4-s + 2·8-s + 8·11-s + 2·16-s + 16·22-s + 4·23-s + 9·25-s + 22·29-s − 12·32-s − 12·37-s − 6·43-s + 40·44-s + 8·46-s + 18·50-s − 56·53-s + 44·58-s − 17·64-s − 76·71-s − 24·74-s + 6·79-s − 12·86-s + 16·88-s + 20·92-s + 45·100-s − 112·106-s − 52·107-s + ⋯
L(s)  = 1  + 1.41·2-s + 5/2·4-s + 0.707·8-s + 2.41·11-s + 1/2·16-s + 3.41·22-s + 0.834·23-s + 9/5·25-s + 4.08·29-s − 2.12·32-s − 1.97·37-s − 0.914·43-s + 6.03·44-s + 1.17·46-s + 2.54·50-s − 7.69·53-s + 5.77·58-s − 2.12·64-s − 9.01·71-s − 2.78·74-s + 0.675·79-s − 1.29·86-s + 1.70·88-s + 2.08·92-s + 9/2·100-s − 10.8·106-s − 5.02·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{36} \cdot 7^{24}\)
Sign: $1$
Analytic conductor: \(1.93216\times 10^{12}\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{36} \cdot 7^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(35.05656959\)
\(L(\frac12)\) \(\approx\) \(35.05656959\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( ( 1 - T - T^{2} + p^{2} T^{3} - 3 T^{4} - p T^{5} + 13 T^{6} - p^{2} T^{7} - 3 p^{2} T^{8} + p^{5} T^{9} - p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \)
5 \( 1 - 9 T^{2} + 18 T^{4} + 131 T^{6} - 747 T^{8} + 738 T^{10} + 3201 T^{12} + 738 p^{2} T^{14} - 747 p^{4} T^{16} + 131 p^{6} T^{18} + 18 p^{8} T^{20} - 9 p^{10} T^{22} + p^{12} T^{24} \)
11 \( ( 1 - 4 T - 16 T^{2} + 46 T^{3} + 324 T^{4} - 452 T^{5} - 2969 T^{6} - 452 p T^{7} + 324 p^{2} T^{8} + 46 p^{3} T^{9} - 16 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
13 \( 1 - 3 p T^{2} + 51 p T^{4} - 6584 T^{6} + 5157 p T^{8} - 102645 p T^{10} + 22407342 T^{12} - 102645 p^{3} T^{14} + 5157 p^{5} T^{16} - 6584 p^{6} T^{18} + 51 p^{9} T^{20} - 3 p^{11} T^{22} + p^{12} T^{24} \)
17 \( ( 1 + 18 T^{2} + 216 T^{4} + 2797 T^{6} + 216 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 + 39 T^{2} + 1011 T^{4} + 20009 T^{6} + 1011 p^{2} T^{8} + 39 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
23 \( ( 1 - 2 T - 40 T^{2} - 22 T^{3} + 858 T^{4} + 1538 T^{5} - 21221 T^{6} + 1538 p T^{7} + 858 p^{2} T^{8} - 22 p^{3} T^{9} - 40 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - 11 T + 20 T^{2} - 13 T^{3} + 1233 T^{4} - 262 T^{5} - 47411 T^{6} - 262 p T^{7} + 1233 p^{2} T^{8} - 13 p^{3} T^{9} + 20 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
31 \( 1 - 57 T^{2} - 678 T^{4} + 7483 T^{6} + 5054805 T^{8} - 2471706 p T^{10} - 2776215 p^{2} T^{12} - 2471706 p^{3} T^{14} + 5054805 p^{4} T^{16} + 7483 p^{6} T^{18} - 678 p^{8} T^{20} - 57 p^{10} T^{22} + p^{12} T^{24} \)
37 \( ( 1 + 3 T + 87 T^{2} + 249 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
41 \( 1 - 84 T^{2} + 2334 T^{4} + 75506 T^{6} - 5470866 T^{8} - 18588942 T^{10} + 7812254391 T^{12} - 18588942 p^{2} T^{14} - 5470866 p^{4} T^{16} + 75506 p^{6} T^{18} + 2334 p^{8} T^{20} - 84 p^{10} T^{22} + p^{12} T^{24} \)
43 \( ( 1 + 3 T - 96 T^{2} - 255 T^{3} + 5655 T^{4} + 8382 T^{5} - 250477 T^{6} + 8382 p T^{7} + 5655 p^{2} T^{8} - 255 p^{3} T^{9} - 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( 1 - 99 T^{2} + 486 T^{4} - 19399 T^{6} + 21504483 T^{8} - 613600884 T^{10} - 14184900399 T^{12} - 613600884 p^{2} T^{14} + 21504483 p^{4} T^{16} - 19399 p^{6} T^{18} + 486 p^{8} T^{20} - 99 p^{10} T^{22} + p^{12} T^{24} \)
53 \( ( 1 + 14 T + 170 T^{2} + 1221 T^{3} + 170 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
59 \( 1 - 171 T^{2} + 10764 T^{4} - 658021 T^{6} + 63198909 T^{8} - 3185695998 T^{10} + 106058651361 T^{12} - 3185695998 p^{2} T^{14} + 63198909 p^{4} T^{16} - 658021 p^{6} T^{18} + 10764 p^{8} T^{20} - 171 p^{10} T^{22} + p^{12} T^{24} \)
61 \( 1 - 102 T^{2} - 2514 T^{4} + 108094 T^{6} + 46094616 T^{8} - 815218740 T^{10} - 153775720821 T^{12} - 815218740 p^{2} T^{14} + 46094616 p^{4} T^{16} + 108094 p^{6} T^{18} - 2514 p^{8} T^{20} - 102 p^{10} T^{22} + p^{12} T^{24} \)
67 \( ( 1 - 90 T^{2} + 706 T^{3} + 2070 T^{4} - 31770 T^{5} + 183435 T^{6} - 31770 p T^{7} + 2070 p^{2} T^{8} + 706 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
71 \( ( 1 + 19 T + 329 T^{2} + 2925 T^{3} + 329 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
73 \( ( 1 + 363 T^{2} + 59331 T^{4} + 5567537 T^{6} + 59331 p^{2} T^{8} + 363 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 3 T - 150 T^{2} + 257 T^{3} + 11619 T^{4} - 1710 T^{5} - 932601 T^{6} - 1710 p T^{7} + 11619 p^{2} T^{8} + 257 p^{3} T^{9} - 150 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( 1 - 270 T^{2} + 31896 T^{4} - 3065782 T^{6} + 318136230 T^{8} - 24401026026 T^{10} + 1632699460815 T^{12} - 24401026026 p^{2} T^{14} + 318136230 p^{4} T^{16} - 3065782 p^{6} T^{18} + 31896 p^{8} T^{20} - 270 p^{10} T^{22} + p^{12} T^{24} \)
89 \( ( 1 + 288 T^{2} + 36990 T^{4} + 3427693 T^{6} + 36990 p^{2} T^{8} + 288 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( 1 - 471 T^{2} + 121776 T^{4} - 22400861 T^{6} + 3247635573 T^{8} - 392084795286 T^{10} + 40704346255641 T^{12} - 392084795286 p^{2} T^{14} + 3247635573 p^{4} T^{16} - 22400861 p^{6} T^{18} + 121776 p^{8} T^{20} - 471 p^{10} T^{22} + p^{12} T^{24} \)
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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

−3.08965919559596933433864444967, −3.03502395574556906621397140748, −2.92127791181208404222520388550, −2.83609422897789860176796927309, −2.75110621242514839724020129125, −2.63556187950415170784297958086, −2.58231977940526144964313869181, −2.45783688330130718072644869211, −2.34142861341577491704929111373, −2.26669987847789043760124362571, −2.08120230561544408292792586359, −1.81221675905534425231295381681, −1.78174862329570053816136551702, −1.74312586546467616082131888707, −1.58358840935837128510513700338, −1.55491300754582599388401451211, −1.34805713631555322319462344366, −1.28147385383388691625179733445, −1.25544276282299860403081880285, −1.24443576818777770188705864344, −1.11289575825172498147000454340, −0.57000232979116910832636912616, −0.42616091194259896210819977014, −0.42144401285893494560546704421, −0.26655574186838148659208416286, 0.26655574186838148659208416286, 0.42144401285893494560546704421, 0.42616091194259896210819977014, 0.57000232979116910832636912616, 1.11289575825172498147000454340, 1.24443576818777770188705864344, 1.25544276282299860403081880285, 1.28147385383388691625179733445, 1.34805713631555322319462344366, 1.55491300754582599388401451211, 1.58358840935837128510513700338, 1.74312586546467616082131888707, 1.78174862329570053816136551702, 1.81221675905534425231295381681, 2.08120230561544408292792586359, 2.26669987847789043760124362571, 2.34142861341577491704929111373, 2.45783688330130718072644869211, 2.58231977940526144964313869181, 2.63556187950415170784297958086, 2.75110621242514839724020129125, 2.83609422897789860176796927309, 2.92127791181208404222520388550, 3.03502395574556906621397140748, 3.08965919559596933433864444967

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

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