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Properties

Label 24-210e12-1.1-c1e12-0-0
Degree $24$
Conductor $7.356\times 10^{27}$
Sign $1$
Analytic cond. $494.261$
Root an. cond. $1.29493$
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  + 3·4-s + 3·9-s − 6·11-s + 3·16-s + 6·19-s − 48·29-s + 9·36-s − 36·41-s − 18·44-s + 12·49-s − 24·59-s − 12·61-s − 2·64-s + 72·71-s + 18·76-s + 48·79-s + 3·81-s − 12·89-s − 18·99-s + 48·101-s + 36·109-s − 144·116-s + 21·121-s + 40·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 3/2·4-s + 9-s − 1.80·11-s + 3/4·16-s + 1.37·19-s − 8.91·29-s + 3/2·36-s − 5.62·41-s − 2.71·44-s + 12/7·49-s − 3.12·59-s − 1.53·61-s − 1/4·64-s + 8.54·71-s + 2.06·76-s + 5.40·79-s + 1/3·81-s − 1.27·89-s − 1.80·99-s + 4.77·101-s + 3.44·109-s − 13.3·116-s + 1.90·121-s + 3.57·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(494.261\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7738699553\)
\(L(\frac12)\) \(\approx\) \(0.7738699553\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{3} \)
3 \( ( 1 - T^{2} + T^{4} )^{3} \)
5 \( ( 1 - 4 p T^{3} + p^{3} T^{6} )^{2} \)
7 \( 1 - 12 T^{2} + 120 T^{4} - 790 T^{6} + 120 p^{2} T^{8} - 12 p^{4} T^{10} + p^{6} T^{12} \)
good11 \( ( 1 + 3 T + 3 T^{2} - 16 T^{3} - 81 T^{4} + 69 T^{5} + 1466 T^{6} + 69 p T^{7} - 81 p^{2} T^{8} - 16 p^{3} T^{9} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
13 \( ( 1 - 45 T^{2} + 1107 T^{4} - 17390 T^{6} + 1107 p^{2} T^{8} - 45 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 - 30 T^{2} - 27 T^{4} + 2790 T^{6} + 132426 T^{8} - 36150 p T^{10} - 142387 p^{2} T^{12} - 36150 p^{3} T^{14} + 132426 p^{4} T^{16} + 2790 p^{6} T^{18} - 27 p^{8} T^{20} - 30 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 - 3 T - 36 T^{2} + 45 T^{3} + 900 T^{4} - 3 T^{5} - 19906 T^{6} - 3 p T^{7} + 900 p^{2} T^{8} + 45 p^{3} T^{9} - 36 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 81 T^{2} + 3582 T^{4} + 90087 T^{6} + 1174464 T^{8} - 7396947 T^{10} - 518277076 T^{12} - 7396947 p^{2} T^{14} + 1174464 p^{4} T^{16} + 90087 p^{6} T^{18} + 3582 p^{8} T^{20} + 81 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 + 12 T + 120 T^{2} + 720 T^{3} + 120 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
31 \( ( 1 - 78 T^{2} - 20 T^{3} + 3666 T^{4} + 780 T^{5} - 128922 T^{6} + 780 p T^{7} + 3666 p^{2} T^{8} - 20 p^{3} T^{9} - 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( 1 + 165 T^{2} + 14838 T^{4} + 911635 T^{6} + 43071816 T^{8} + 1725566145 T^{10} + 64531645932 T^{12} + 1725566145 p^{2} T^{14} + 43071816 p^{4} T^{16} + 911635 p^{6} T^{18} + 14838 p^{8} T^{20} + 165 p^{10} T^{22} + p^{12} T^{24} \)
41 \( ( 1 + 9 T + 75 T^{2} + 610 T^{3} + 75 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
43 \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{6} \)
47 \( 1 + 9 T^{2} - 1578 T^{4} - 306297 T^{6} - 2357976 T^{8} + 244410597 T^{10} + 40880244764 T^{12} + 244410597 p^{2} T^{14} - 2357976 p^{4} T^{16} - 306297 p^{6} T^{18} - 1578 p^{8} T^{20} + 9 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 + 51 T^{2} - 6213 T^{4} - 119588 T^{6} + 38415969 T^{8} + 385077393 T^{10} - 113446795386 T^{12} + 385077393 p^{2} T^{14} + 38415969 p^{4} T^{16} - 119588 p^{6} T^{18} - 6213 p^{8} T^{20} + 51 p^{10} T^{22} + p^{12} T^{24} \)
59 \( ( 1 + 12 T - 6 T^{2} - 160 T^{3} + 1890 T^{4} - 23628 T^{5} - 457606 T^{6} - 23628 p T^{7} + 1890 p^{2} T^{8} - 160 p^{3} T^{9} - 6 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 6 T - 39 T^{2} + 410 T^{3} + 930 T^{4} - 31074 T^{5} - 22719 T^{6} - 31074 p T^{7} + 930 p^{2} T^{8} + 410 p^{3} T^{9} - 39 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 150 T^{2} + 6333 T^{4} - 110030 T^{6} - 1332774 T^{8} + 1742888430 T^{10} + 160875783717 T^{12} + 1742888430 p^{2} T^{14} - 1332774 p^{4} T^{16} - 110030 p^{6} T^{18} + 6333 p^{8} T^{20} + 150 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 - 6 T + p T^{2} )^{12} \)
73 \( ( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} )^{3} \)
79 \( ( 1 - 24 T + 162 T^{2} - 1548 T^{3} + 38034 T^{4} - 316212 T^{5} + 1440614 T^{6} - 316212 p T^{7} + 38034 p^{2} T^{8} - 1548 p^{3} T^{9} + 162 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( ( 1 - 300 T^{2} + 40392 T^{4} - 3707850 T^{6} + 40392 p^{2} T^{8} - 300 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 + 6 T - 123 T^{2} - 398 T^{3} + 7074 T^{4} - 14802 T^{5} - 668251 T^{6} - 14802 p T^{7} + 7074 p^{2} T^{8} - 398 p^{3} T^{9} - 123 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( ( 1 - 240 T^{2} + 16752 T^{4} - 546370 T^{6} + 16752 p^{2} T^{8} - 240 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
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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

−4.54322169588407932818350892413, −4.01737986502264136138026553278, −4.01173288424606710182633207537, −3.89503626144813720107609365513, −3.63925073547285117400037583568, −3.63262841114390890811497598370, −3.61916863167524711020739878430, −3.52360725090433845882125041302, −3.50593206615089233627659328247, −3.25119443692124351327819622914, −3.15768973355124042793576692949, −3.14408311242925081863406971467, −2.99549560776680708503235728872, −2.58878454601994060807560626861, −2.29474100624001875363625903247, −2.23008101522047363477640887926, −2.18094162019118144585178366430, −2.15010600216965396995730742929, −1.90260332856618686789969354837, −1.83807705045679515242869731728, −1.73091716192057246190613261842, −1.72127552125425944477238926834, −1.02527931871890960683364851011, −0.956375160310457147250412355678, −0.20784542283062059267330616768, 0.20784542283062059267330616768, 0.956375160310457147250412355678, 1.02527931871890960683364851011, 1.72127552125425944477238926834, 1.73091716192057246190613261842, 1.83807705045679515242869731728, 1.90260332856618686789969354837, 2.15010600216965396995730742929, 2.18094162019118144585178366430, 2.23008101522047363477640887926, 2.29474100624001875363625903247, 2.58878454601994060807560626861, 2.99549560776680708503235728872, 3.14408311242925081863406971467, 3.15768973355124042793576692949, 3.25119443692124351327819622914, 3.50593206615089233627659328247, 3.52360725090433845882125041302, 3.61916863167524711020739878430, 3.63262841114390890811497598370, 3.63925073547285117400037583568, 3.89503626144813720107609365513, 4.01173288424606710182633207537, 4.01737986502264136138026553278, 4.54322169588407932818350892413

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

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