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Properties

Label 12-3344e6-1.1-c1e6-0-2
Degree $12$
Conductor $1.398\times 10^{21}$
Sign $1$
Analytic cond. $3.62460\times 10^{8}$
Root an. cond. $5.16739$
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  − 3-s + 5·5-s + 2·7-s + 6·11-s + 8·13-s − 5·15-s − 2·17-s + 6·19-s − 2·21-s − 5·23-s + 4·25-s − 27-s + 10·29-s − 3·31-s − 6·33-s + 10·35-s + 7·37-s − 8·39-s + 6·41-s − 16·43-s − 13·49-s + 2·51-s + 20·53-s + 30·55-s − 6·57-s − 15·59-s + 24·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 2.23·5-s + 0.755·7-s + 1.80·11-s + 2.21·13-s − 1.29·15-s − 0.485·17-s + 1.37·19-s − 0.436·21-s − 1.04·23-s + 4/5·25-s − 0.192·27-s + 1.85·29-s − 0.538·31-s − 1.04·33-s + 1.69·35-s + 1.15·37-s − 1.28·39-s + 0.937·41-s − 2.43·43-s − 1.85·49-s + 0.280·51-s + 2.74·53-s + 4.04·55-s − 0.794·57-s − 1.95·59-s + 3.07·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 11^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(3.62460\times 10^{8}\)
Root analytic conductor: \(5.16739\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 11^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(20.23993358\)
\(L(\frac12)\) \(\approx\) \(20.23993358\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( ( 1 - T )^{6} \)
19 \( ( 1 - T )^{6} \)
good3 \( 1 + T + T^{2} + 2 T^{3} - 2 p T^{5} + 2 p T^{6} - 2 p^{2} T^{7} + 2 p^{3} T^{9} + p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 - p T + 21 T^{2} - 48 T^{3} + 96 T^{4} - 86 T^{5} + 178 T^{6} - 86 p T^{7} + 96 p^{2} T^{8} - 48 p^{3} T^{9} + 21 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
7 \( 1 - 2 T + 17 T^{2} - 12 T^{3} + 116 T^{4} + 82 T^{5} + 548 T^{6} + 82 p T^{7} + 116 p^{2} T^{8} - 12 p^{3} T^{9} + 17 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 8 T + 55 T^{2} - 274 T^{3} + 1148 T^{4} - 4180 T^{5} + 15612 T^{6} - 4180 p T^{7} + 1148 p^{2} T^{8} - 274 p^{3} T^{9} + 55 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 2 T + 30 T^{2} + 122 T^{3} + 607 T^{4} + 2084 T^{5} + 12868 T^{6} + 2084 p T^{7} + 607 p^{2} T^{8} + 122 p^{3} T^{9} + 30 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 5 T + 50 T^{2} + 243 T^{3} + 1967 T^{4} + 8550 T^{5} + 54684 T^{6} + 8550 p T^{7} + 1967 p^{2} T^{8} + 243 p^{3} T^{9} + 50 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 10 T + 83 T^{2} - 422 T^{3} + 2572 T^{4} - 8554 T^{5} + 46924 T^{6} - 8554 p T^{7} + 2572 p^{2} T^{8} - 422 p^{3} T^{9} + 83 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 3 T + 107 T^{2} + 78 T^{3} + 4418 T^{4} - 6464 T^{5} + 127702 T^{6} - 6464 p T^{7} + 4418 p^{2} T^{8} + 78 p^{3} T^{9} + 107 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 7 T + 112 T^{2} - 967 T^{3} + 8191 T^{4} - 54478 T^{5} + 394928 T^{6} - 54478 p T^{7} + 8191 p^{2} T^{8} - 967 p^{3} T^{9} + 112 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 6 T + 203 T^{2} - 766 T^{3} + 16684 T^{4} - 41824 T^{5} + 822548 T^{6} - 41824 p T^{7} + 16684 p^{2} T^{8} - 766 p^{3} T^{9} + 203 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 16 T + 209 T^{2} + 1924 T^{3} + 18072 T^{4} + 134610 T^{5} + 980204 T^{6} + 134610 p T^{7} + 18072 p^{2} T^{8} + 1924 p^{3} T^{9} + 209 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 158 T^{2} - 176 T^{3} + 13039 T^{4} - 13104 T^{5} + 746020 T^{6} - 13104 p T^{7} + 13039 p^{2} T^{8} - 176 p^{3} T^{9} + 158 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 20 T + 370 T^{2} - 4436 T^{3} + 49815 T^{4} - 434728 T^{5} + 3532444 T^{6} - 434728 p T^{7} + 49815 p^{2} T^{8} - 4436 p^{3} T^{9} + 370 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 15 T + 248 T^{2} + 3029 T^{3} + 31879 T^{4} + 292034 T^{5} + 2417840 T^{6} + 292034 p T^{7} + 31879 p^{2} T^{8} + 3029 p^{3} T^{9} + 248 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 24 T + 426 T^{2} - 5624 T^{3} + 63015 T^{4} - 601104 T^{5} + 4996652 T^{6} - 601104 p T^{7} + 63015 p^{2} T^{8} - 5624 p^{3} T^{9} + 426 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 25 T + 479 T^{2} + 5946 T^{3} + 65282 T^{4} + 570676 T^{5} + 4992830 T^{6} + 570676 p T^{7} + 65282 p^{2} T^{8} + 5946 p^{3} T^{9} + 479 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 9 T + 269 T^{2} - 1862 T^{3} + 35908 T^{4} - 199430 T^{5} + 3020546 T^{6} - 199430 p T^{7} + 35908 p^{2} T^{8} - 1862 p^{3} T^{9} + 269 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 26 T + 458 T^{2} + 6474 T^{3} + 1031 p T^{4} + 760652 T^{5} + 6945740 T^{6} + 760652 p T^{7} + 1031 p^{3} T^{8} + 6474 p^{3} T^{9} + 458 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 16 T + 454 T^{2} - 5264 T^{3} + 82511 T^{4} - 740480 T^{5} + 8351892 T^{6} - 740480 p T^{7} + 82511 p^{2} T^{8} - 5264 p^{3} T^{9} + 454 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 2 T + 421 T^{2} - 744 T^{3} + 79316 T^{4} - 117434 T^{5} + 102428 p T^{6} - 117434 p T^{7} + 79316 p^{2} T^{8} - 744 p^{3} T^{9} + 421 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 7 T + 308 T^{2} - 1931 T^{3} + 43111 T^{4} - 259078 T^{5} + 4205512 T^{6} - 259078 p T^{7} + 43111 p^{2} T^{8} - 1931 p^{3} T^{9} + 308 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 37 T + 916 T^{2} - 16581 T^{3} + 246255 T^{4} - 3080646 T^{5} + 32767192 T^{6} - 3080646 p T^{7} + 246255 p^{2} T^{8} - 16581 p^{3} T^{9} + 916 p^{4} T^{10} - 37 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.56891039344874644803739881293, −4.35243355689214477204737166347, −4.12396708757014634792821579990, −3.98090417977121357932676660361, −3.90296798119748229541168374277, −3.75424567039116553845107986065, −3.56543608520028477300660359785, −3.42814910990862365183226094700, −3.28638151726355241030223251800, −3.12729999421482292703132001442, −2.95623861908621604452654817825, −2.81537084908872564234663877899, −2.76082435602366985662187838653, −2.24448104569691121917397665888, −2.03310294096895119555916590877, −2.01482459983854826048500665619, −1.94699582896425940345597968448, −1.79915736880155457527395826386, −1.65086538954313524090494179939, −1.56096570250052405401613788032, −1.02374187244259494323978266745, −0.943616483952674192835814904882, −0.920825191983100859628425416162, −0.72428638465717122638185981304, −0.30521649179204643516154821287, 0.30521649179204643516154821287, 0.72428638465717122638185981304, 0.920825191983100859628425416162, 0.943616483952674192835814904882, 1.02374187244259494323978266745, 1.56096570250052405401613788032, 1.65086538954313524090494179939, 1.79915736880155457527395826386, 1.94699582896425940345597968448, 2.01482459983854826048500665619, 2.03310294096895119555916590877, 2.24448104569691121917397665888, 2.76082435602366985662187838653, 2.81537084908872564234663877899, 2.95623861908621604452654817825, 3.12729999421482292703132001442, 3.28638151726355241030223251800, 3.42814910990862365183226094700, 3.56543608520028477300660359785, 3.75424567039116553845107986065, 3.90296798119748229541168374277, 3.98090417977121357932676660361, 4.12396708757014634792821579990, 4.35243355689214477204737166347, 4.56891039344874644803739881293

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

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