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Properties

Label 24-189e12-1.1-c1e12-0-1
Degree $24$
Conductor $2.078\times 10^{27}$
Sign $1$
Analytic cond. $139.594$
Root an. cond. $1.22848$
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  − 2·4-s − 8·7-s + 5·16-s − 6·19-s + 3·25-s + 16·28-s − 12·31-s + 8·37-s + 20·43-s + 44·49-s + 18·61-s + 36·67-s − 42·73-s + 12·76-s − 36·79-s − 6·100-s − 30·103-s + 20·109-s − 40·112-s − 29·121-s + 24·124-s + 127-s + 131-s + 48·133-s + 137-s + 139-s − 16·148-s + ⋯
L(s)  = 1  − 4-s − 3.02·7-s + 5/4·16-s − 1.37·19-s + 3/5·25-s + 3.02·28-s − 2.15·31-s + 1.31·37-s + 3.04·43-s + 44/7·49-s + 2.30·61-s + 4.39·67-s − 4.91·73-s + 1.37·76-s − 4.05·79-s − 3/5·100-s − 2.95·103-s + 1.91·109-s − 3.77·112-s − 2.63·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 4.16·133-s + 0.0854·137-s + 0.0848·139-s − 1.31·148-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(3^{36} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(139.594\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{36} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4613696266\)
\(L(\frac12)\) \(\approx\) \(0.4613696266\)
\(L(\frac{3}{2})\) 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 \( ( 1 + 4 T + 2 T^{2} - 11 T^{3} + 2 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
good2 \( 1 + p T^{2} - T^{4} - 3 p^{2} T^{6} - 17 T^{8} + 7 T^{10} + 85 T^{12} + 7 p^{2} T^{14} - 17 p^{4} T^{16} - 3 p^{8} T^{18} - p^{8} T^{20} + p^{11} T^{22} + p^{12} T^{24} \)
5 \( 1 - 3 T^{2} - 24 T^{4} + 11 p^{2} T^{6} - 297 T^{8} - 3096 T^{10} + 33369 T^{12} - 3096 p^{2} T^{14} - 297 p^{4} T^{16} + 11 p^{8} T^{18} - 24 p^{8} T^{20} - 3 p^{10} T^{22} + p^{12} T^{24} \)
11 \( 1 + 29 T^{2} + 40 p T^{4} + 3147 T^{6} - 10709 T^{8} - 688988 T^{10} - 10240799 T^{12} - 688988 p^{2} T^{14} - 10709 p^{4} T^{16} + 3147 p^{6} T^{18} + 40 p^{9} T^{20} + 29 p^{10} T^{22} + p^{12} T^{24} \)
13 \( ( 1 - 36 T^{2} + 720 T^{4} - 10271 T^{6} + 720 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 - 72 T^{2} + 2664 T^{4} - 74146 T^{6} + 1759536 T^{8} - 36331632 T^{10} + 658037283 T^{12} - 36331632 p^{2} T^{14} + 1759536 p^{4} T^{16} - 74146 p^{6} T^{18} + 2664 p^{8} T^{20} - 72 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 + 3 T + 36 T^{2} + 99 T^{3} + 549 T^{4} + 1788 T^{5} + 8413 T^{6} + 1788 p T^{7} + 549 p^{2} T^{8} + 99 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 44 T^{2} + 2132 T^{4} + 55302 T^{6} + 1548472 T^{8} + 27701944 T^{10} + 755610907 T^{12} + 27701944 p^{2} T^{14} + 1548472 p^{4} T^{16} + 55302 p^{6} T^{18} + 2132 p^{8} T^{20} + 44 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 - 137 T^{2} + 8645 T^{4} - 319673 T^{6} + 8645 p^{2} T^{8} - 137 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 + 6 T + 42 T^{2} + 180 T^{3} - 168 T^{4} - 7836 T^{5} - 41627 T^{6} - 7836 p T^{7} - 168 p^{2} T^{8} + 180 p^{3} T^{9} + 42 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 4 T - 76 T^{2} + 90 T^{3} + 112 p T^{4} + 688 T^{5} - 181325 T^{6} + 688 p T^{7} + 112 p^{3} T^{8} + 90 p^{3} T^{9} - 76 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( ( 1 + 132 T^{2} + 8472 T^{4} + 388519 T^{6} + 8472 p^{2} T^{8} + 132 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 - 5 T + 113 T^{2} - 431 T^{3} + 113 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
47 \( 1 - 147 T^{2} + 9984 T^{4} - 446341 T^{6} + 16664751 T^{8} - 505982232 T^{10} + 16085788593 T^{12} - 505982232 p^{2} T^{14} + 16664751 p^{4} T^{16} - 446341 p^{6} T^{18} + 9984 p^{8} T^{20} - 147 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 + 200 T^{2} + 21164 T^{4} + 1425678 T^{6} + 67732372 T^{8} + 2466420796 T^{10} + 102771938515 T^{12} + 2466420796 p^{2} T^{14} + 67732372 p^{4} T^{16} + 1425678 p^{6} T^{18} + 21164 p^{8} T^{20} + 200 p^{10} T^{22} + p^{12} T^{24} \)
59 \( 1 - 291 T^{2} + 46596 T^{4} - 5329417 T^{6} + 480785571 T^{8} - 35925708996 T^{10} + 2284341412497 T^{12} - 35925708996 p^{2} T^{14} + 480785571 p^{4} T^{16} - 5329417 p^{6} T^{18} + 46596 p^{8} T^{20} - 291 p^{10} T^{22} + p^{12} T^{24} \)
61 \( ( 1 - 9 T + 159 T^{2} - 1188 T^{3} + 11505 T^{4} - 60363 T^{5} + 676942 T^{6} - 60363 p T^{7} + 11505 p^{2} T^{8} - 1188 p^{3} T^{9} + 159 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 18 T + 108 T^{2} - 418 T^{3} + 954 T^{4} + 65430 T^{5} - 968637 T^{6} + 65430 p T^{7} + 954 p^{2} T^{8} - 418 p^{3} T^{9} + 108 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 341 T^{2} + 53765 T^{4} - 4892609 T^{6} + 53765 p^{2} T^{8} - 341 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 + 21 T + 342 T^{2} + 4095 T^{3} + 40149 T^{4} + 343806 T^{5} + 3034951 T^{6} + 343806 p T^{7} + 40149 p^{2} T^{8} + 4095 p^{3} T^{9} + 342 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
79 \( ( 1 + 18 T + 72 T^{2} - 1138 T^{3} - 7470 T^{4} + 110574 T^{5} + 1995747 T^{6} + 110574 p T^{7} - 7470 p^{2} T^{8} - 1138 p^{3} T^{9} + 72 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( ( 1 + 339 T^{2} + 56361 T^{4} + 5822683 T^{6} + 56361 p^{2} T^{8} + 339 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( 1 - 333 T^{2} + 53955 T^{4} - 6596860 T^{6} + 732206169 T^{8} - 72458005383 T^{10} + 6566534648358 T^{12} - 72458005383 p^{2} T^{14} + 732206169 p^{4} T^{16} - 6596860 p^{6} T^{18} + 53955 p^{8} T^{20} - 333 p^{10} T^{22} + p^{12} T^{24} \)
97 \( ( 1 - 378 T^{2} + 65727 T^{4} - 7461452 T^{6} + 65727 p^{2} T^{8} - 378 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
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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

−4.35668515324173356660748074740, −4.27659345897763331557502614994, −4.02667237737883887843363490922, −4.01017293493023535629077665395, −3.95245326167322305843080169927, −3.93523346525568828505450698527, −3.86599776265850199085720843752, −3.69830741621834877958108922721, −3.51499434626570847547450702740, −3.37499441126844142504108111849, −3.17840961417834810906292785729, −2.96794080793941839423489212343, −2.87892996628006335990053182938, −2.77565145304062586900322324307, −2.73381909199278755860885802027, −2.52578972721512110181354071767, −2.44542988429271814643989586312, −2.40723375473541433623573699468, −2.03115188513940804053021374083, −1.74969618381639859053982436326, −1.63649846551935434181554606907, −1.28646409251738375091327508019, −1.10171270149085016099708516194, −0.69728505514162076566055727220, −0.31040621761631133695620562838, 0.31040621761631133695620562838, 0.69728505514162076566055727220, 1.10171270149085016099708516194, 1.28646409251738375091327508019, 1.63649846551935434181554606907, 1.74969618381639859053982436326, 2.03115188513940804053021374083, 2.40723375473541433623573699468, 2.44542988429271814643989586312, 2.52578972721512110181354071767, 2.73381909199278755860885802027, 2.77565145304062586900322324307, 2.87892996628006335990053182938, 2.96794080793941839423489212343, 3.17840961417834810906292785729, 3.37499441126844142504108111849, 3.51499434626570847547450702740, 3.69830741621834877958108922721, 3.86599776265850199085720843752, 3.93523346525568828505450698527, 3.95245326167322305843080169927, 4.01017293493023535629077665395, 4.02667237737883887843363490922, 4.27659345897763331557502614994, 4.35668515324173356660748074740

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

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