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Properties

Label 12-304e6-1.1-c1e6-0-6
Degree $12$
Conductor $7.893\times 10^{14}$
Sign $1$
Analytic cond. $204.599$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s − 6·5-s − 6·7-s + 36·9-s − 6·11-s − 15·13-s + 54·15-s − 9·17-s − 18·19-s + 54·21-s − 18·23-s + 18·25-s − 80·27-s − 3·29-s + 3·31-s + 54·33-s + 36·35-s − 24·37-s + 135·39-s − 15·41-s + 21·43-s − 216·45-s − 21·47-s + 30·49-s + 81·51-s − 27·53-s + 36·55-s + ⋯
L(s)  = 1  − 5.19·3-s − 2.68·5-s − 2.26·7-s + 12·9-s − 1.80·11-s − 4.16·13-s + 13.9·15-s − 2.18·17-s − 4.12·19-s + 11.7·21-s − 3.75·23-s + 18/5·25-s − 15.3·27-s − 0.557·29-s + 0.538·31-s + 9.40·33-s + 6.08·35-s − 3.94·37-s + 21.6·39-s − 2.34·41-s + 3.20·43-s − 32.1·45-s − 3.06·47-s + 30/7·49-s + 11.3·51-s − 3.70·53-s + 4.85·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(204.599\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{24} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + 18 T + 144 T^{2} + 737 T^{3} + 144 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( 1 + p^{2} T + 5 p^{2} T^{2} + 161 T^{3} + 50 p^{2} T^{4} + 38 p^{3} T^{5} + 1945 T^{6} + 38 p^{4} T^{7} + 50 p^{4} T^{8} + 161 p^{3} T^{9} + 5 p^{6} T^{10} + p^{7} T^{11} + p^{6} T^{12} \)
5 \( 1 + 6 T + 18 T^{2} + 9 p T^{3} + 81 T^{4} + 87 T^{5} + 109 T^{6} + 87 p T^{7} + 81 p^{2} T^{8} + 9 p^{4} T^{9} + 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 6 T + 6 T^{2} + 6 T^{3} + 24 p T^{4} + 384 T^{5} + 191 T^{6} + 384 p T^{7} + 24 p^{3} T^{8} + 6 p^{3} T^{9} + 6 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 6 T - 6 T^{2} - 14 T^{3} + 504 T^{4} + 768 T^{5} - 3409 T^{6} + 768 p T^{7} + 504 p^{2} T^{8} - 14 p^{3} T^{9} - 6 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 15 T + 96 T^{2} + 298 T^{3} - 9 p T^{4} - 6255 T^{5} - 32679 T^{6} - 6255 p T^{7} - 9 p^{3} T^{8} + 298 p^{3} T^{9} + 96 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 9 T + 36 T^{2} + 40 T^{3} - 369 T^{4} - 2673 T^{5} - 10927 T^{6} - 2673 p T^{7} - 369 p^{2} T^{8} + 40 p^{3} T^{9} + 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 18 T + 180 T^{2} + 1354 T^{3} + 8784 T^{4} + 49464 T^{5} + 249209 T^{6} + 49464 p T^{7} + 8784 p^{2} T^{8} + 1354 p^{3} T^{9} + 180 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 36 T^{2} + 378 T^{3} + 1872 T^{4} + 9921 T^{5} + 94159 T^{6} + 9921 p T^{7} + 1872 p^{2} T^{8} + 378 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 3 T - 60 T^{2} + 59 T^{3} + 2223 T^{4} + 774 T^{5} - 78969 T^{6} + 774 p T^{7} + 2223 p^{2} T^{8} + 59 p^{3} T^{9} - 60 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( ( 1 + 12 T + 150 T^{2} + 907 T^{3} + 150 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 15 T + 150 T^{2} + 1000 T^{3} + 8025 T^{4} + 61965 T^{5} + 472529 T^{6} + 61965 p T^{7} + 8025 p^{2} T^{8} + 1000 p^{3} T^{9} + 150 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 21 T + 174 T^{2} - 454 T^{3} - 3879 T^{4} + 58707 T^{5} - 460263 T^{6} + 58707 p T^{7} - 3879 p^{2} T^{8} - 454 p^{3} T^{9} + 174 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 21 T + 186 T^{2} + 908 T^{3} + 2715 T^{4} + 657 T^{5} - 54025 T^{6} + 657 p T^{7} + 2715 p^{2} T^{8} + 908 p^{3} T^{9} + 186 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 27 T + 324 T^{2} + 2178 T^{3} + 11259 T^{4} + 87453 T^{5} + 753949 T^{6} + 87453 p T^{7} + 11259 p^{2} T^{8} + 2178 p^{3} T^{9} + 324 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 6 T + 6 T^{2} + 332 T^{3} - 2208 T^{4} - 10908 T^{5} + 340589 T^{6} - 10908 p T^{7} - 2208 p^{2} T^{8} + 332 p^{3} T^{9} + 6 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 12 T - 24 T^{2} + 1238 T^{3} - 7956 T^{4} - 48996 T^{5} + 978639 T^{6} - 48996 p T^{7} - 7956 p^{2} T^{8} + 1238 p^{3} T^{9} - 24 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 18 T + 252 T^{2} + 1838 T^{3} + 4968 T^{4} - 64692 T^{5} - 990987 T^{6} - 64692 p T^{7} + 4968 p^{2} T^{8} + 1838 p^{3} T^{9} + 252 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 18 T + 252 T^{2} + 1814 T^{3} + 5832 T^{4} - 54324 T^{5} - 887815 T^{6} - 54324 p T^{7} + 5832 p^{2} T^{8} + 1814 p^{3} T^{9} + 252 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 36 T + 576 T^{2} - 5760 T^{3} + 51984 T^{4} - 544320 T^{5} + 5272343 T^{6} - 544320 p T^{7} + 51984 p^{2} T^{8} - 5760 p^{3} T^{9} + 576 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 27 T + 180 T^{2} - 1964 T^{3} - 32490 T^{4} - 11547 T^{5} + 1937037 T^{6} - 11547 p T^{7} - 32490 p^{2} T^{8} - 1964 p^{3} T^{9} + 180 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 12 T - 150 T^{2} - 562 T^{3} + 37494 T^{4} + 98250 T^{5} - 2840605 T^{6} + 98250 p T^{7} + 37494 p^{2} T^{8} - 562 p^{3} T^{9} - 150 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 12 T + 54 T^{2} + 1035 T^{3} - 279 T^{4} - 69891 T^{5} + 3961 T^{6} - 69891 p T^{7} - 279 p^{2} T^{8} + 1035 p^{3} T^{9} + 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 18 T + 270 T^{2} + 4178 T^{3} + 50940 T^{4} + 535752 T^{5} + 5820555 T^{6} + 535752 p T^{7} + 50940 p^{2} T^{8} + 4178 p^{3} T^{9} + 270 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.00820039279161874905501137167, −6.49971060897362979511561371440, −6.38134130913356182863179587099, −6.34543112826713143912387064344, −6.28060821305869572482434432607, −6.21765520560200267778015873485, −5.99103586538089416765680574736, −5.54597359423337706076743522041, −5.39594318982216479586036883501, −5.19977406470742622286100624982, −5.19040623208105583996420403525, −5.15433618110883852317273453322, −4.84450292400711764204908889079, −4.43746854256803047399051037185, −4.29341750782648339354552986395, −4.23696712306679260547063002140, −4.20644719544482949991517078395, −3.97297030795175168094658397943, −3.43834989136773901717483753778, −3.25857511079660822708767588263, −2.84908086618182205097568939456, −2.68618014787347857699935654435, −2.24339989875866482874873369859, −2.03519893992654500254372132537, −1.95867123665135618429143645799, 0, 0, 0, 0, 0, 0, 1.95867123665135618429143645799, 2.03519893992654500254372132537, 2.24339989875866482874873369859, 2.68618014787347857699935654435, 2.84908086618182205097568939456, 3.25857511079660822708767588263, 3.43834989136773901717483753778, 3.97297030795175168094658397943, 4.20644719544482949991517078395, 4.23696712306679260547063002140, 4.29341750782648339354552986395, 4.43746854256803047399051037185, 4.84450292400711764204908889079, 5.15433618110883852317273453322, 5.19040623208105583996420403525, 5.19977406470742622286100624982, 5.39594318982216479586036883501, 5.54597359423337706076743522041, 5.99103586538089416765680574736, 6.21765520560200267778015873485, 6.28060821305869572482434432607, 6.34543112826713143912387064344, 6.38134130913356182863179587099, 6.49971060897362979511561371440, 7.00820039279161874905501137167

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

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