Properties

Label 12-912e6-1.1-c5e6-0-0
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $9.79337\times 10^{12}$
Root an. cond. $12.0942$
Motivic weight $5$
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  + 54·3-s − 65·5-s + 149·7-s + 1.70e3·9-s + 203·11-s − 298·13-s − 3.51e3·15-s + 1.31e3·17-s − 2.16e3·19-s + 8.04e3·21-s − 1.23e3·23-s − 4.96e3·25-s + 4.08e4·27-s + 7.35e3·29-s − 1.63e3·31-s + 1.09e4·33-s − 9.68e3·35-s + 1.42e4·37-s − 1.60e4·39-s + 1.47e4·41-s + 4.69e3·43-s − 1.10e5·45-s + 1.09e4·47-s − 2.00e4·49-s + 7.12e4·51-s + 4.75e4·53-s − 1.31e4·55-s + ⋯
L(s)  = 1  + 3.46·3-s − 1.16·5-s + 1.14·7-s + 7·9-s + 0.505·11-s − 0.489·13-s − 4.02·15-s + 1.10·17-s − 1.37·19-s + 3.98·21-s − 0.486·23-s − 1.58·25-s + 10.7·27-s + 1.62·29-s − 0.305·31-s + 1.75·33-s − 1.33·35-s + 1.70·37-s − 1.69·39-s + 1.36·41-s + 0.387·43-s − 8.13·45-s + 0.723·47-s − 1.19·49-s + 3.83·51-s + 2.32·53-s − 0.588·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(9.79337\times 10^{12}\)
Root analytic conductor: \(12.0942\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(145.9494650\)
\(L(\frac12)\) \(\approx\) \(145.9494650\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - p^{2} T )^{6} \)
19 \( ( 1 + p^{2} T )^{6} \)
good5 \( 1 + 13 p T + 9193 T^{2} + 570048 T^{3} + 56551083 T^{4} + 2772728087 T^{5} + 208371213398 T^{6} + 2772728087 p^{5} T^{7} + 56551083 p^{10} T^{8} + 570048 p^{15} T^{9} + 9193 p^{20} T^{10} + 13 p^{26} T^{11} + p^{30} T^{12} \)
7 \( 1 - 149 T + 42241 T^{2} - 4902846 T^{3} + 914468983 T^{4} - 83804005757 T^{5} + 16226188414670 T^{6} - 83804005757 p^{5} T^{7} + 914468983 p^{10} T^{8} - 4902846 p^{15} T^{9} + 42241 p^{20} T^{10} - 149 p^{25} T^{11} + p^{30} T^{12} \)
11 \( 1 - 203 T + 714649 T^{2} - 180600366 T^{3} + 230974642911 T^{4} - 60663268943599 T^{5} + 45681219617264238 T^{6} - 60663268943599 p^{5} T^{7} + 230974642911 p^{10} T^{8} - 180600366 p^{15} T^{9} + 714649 p^{20} T^{10} - 203 p^{25} T^{11} + p^{30} T^{12} \)
13 \( 1 + 298 T + 920718 T^{2} + 209456986 T^{3} + 391499538647 T^{4} + 43114142845012 T^{5} + 132527665855438020 T^{6} + 43114142845012 p^{5} T^{7} + 391499538647 p^{10} T^{8} + 209456986 p^{15} T^{9} + 920718 p^{20} T^{10} + 298 p^{25} T^{11} + p^{30} T^{12} \)
17 \( 1 - 1319 T + 3732191 T^{2} - 4419699844 T^{3} + 7598441400941 T^{4} - 7227337647189717 T^{5} + 12366581643529631462 T^{6} - 7227337647189717 p^{5} T^{7} + 7598441400941 p^{10} T^{8} - 4419699844 p^{15} T^{9} + 3732191 p^{20} T^{10} - 1319 p^{25} T^{11} + p^{30} T^{12} \)
23 \( 1 + 1234 T + 21970906 T^{2} + 12871781846 T^{3} + 218910656120703 T^{4} + 38682222896422964 T^{5} + \)\(15\!\cdots\!28\)\( T^{6} + 38682222896422964 p^{5} T^{7} + 218910656120703 p^{10} T^{8} + 12871781846 p^{15} T^{9} + 21970906 p^{20} T^{10} + 1234 p^{25} T^{11} + p^{30} T^{12} \)
29 \( 1 - 7356 T + 59908074 T^{2} - 284191917532 T^{3} + 1375784374793031 T^{4} - 7811982406089015096 T^{5} + \)\(31\!\cdots\!36\)\( T^{6} - 7811982406089015096 p^{5} T^{7} + 1375784374793031 p^{10} T^{8} - 284191917532 p^{15} T^{9} + 59908074 p^{20} T^{10} - 7356 p^{25} T^{11} + p^{30} T^{12} \)
31 \( 1 + 1632 T + 82566746 T^{2} - 169340216880 T^{3} + 1927430931707455 T^{4} - 21211221201419872272 T^{5} + \)\(21\!\cdots\!36\)\( T^{6} - 21211221201419872272 p^{5} T^{7} + 1927430931707455 p^{10} T^{8} - 169340216880 p^{15} T^{9} + 82566746 p^{20} T^{10} + 1632 p^{25} T^{11} + p^{30} T^{12} \)
37 \( 1 - 14204 T + 263876202 T^{2} - 64196193628 p T^{3} + 25977829825786055 T^{4} - \)\(17\!\cdots\!76\)\( T^{5} + \)\(17\!\cdots\!60\)\( T^{6} - \)\(17\!\cdots\!76\)\( p^{5} T^{7} + 25977829825786055 p^{10} T^{8} - 64196193628 p^{16} T^{9} + 263876202 p^{20} T^{10} - 14204 p^{25} T^{11} + p^{30} T^{12} \)
41 \( 1 - 14734 T + 476141234 T^{2} - 5013722220926 T^{3} + 104146765238939951 T^{4} - \)\(89\!\cdots\!60\)\( T^{5} + \)\(14\!\cdots\!68\)\( T^{6} - \)\(89\!\cdots\!60\)\( p^{5} T^{7} + 104146765238939951 p^{10} T^{8} - 5013722220926 p^{15} T^{9} + 476141234 p^{20} T^{10} - 14734 p^{25} T^{11} + p^{30} T^{12} \)
43 \( 1 - 4693 T + 760626257 T^{2} - 3436506586578 T^{3} + 255080933201428951 T^{4} - \)\(10\!\cdots\!17\)\( T^{5} + \)\(48\!\cdots\!54\)\( T^{6} - \)\(10\!\cdots\!17\)\( p^{5} T^{7} + 255080933201428951 p^{10} T^{8} - 3436506586578 p^{15} T^{9} + 760626257 p^{20} T^{10} - 4693 p^{25} T^{11} + p^{30} T^{12} \)
47 \( 1 - 10955 T + 500585003 T^{2} - 502758028506 T^{3} + 94997854778560581 T^{4} + \)\(58\!\cdots\!73\)\( T^{5} + \)\(21\!\cdots\!54\)\( T^{6} + \)\(58\!\cdots\!73\)\( p^{5} T^{7} + 94997854778560581 p^{10} T^{8} - 502758028506 p^{15} T^{9} + 500585003 p^{20} T^{10} - 10955 p^{25} T^{11} + p^{30} T^{12} \)
53 \( 1 - 47500 T + 1146261594 T^{2} - 22910390554828 T^{3} + 315996988336990935 T^{4} - \)\(40\!\cdots\!52\)\( T^{5} + \)\(95\!\cdots\!24\)\( T^{6} - \)\(40\!\cdots\!52\)\( p^{5} T^{7} + 315996988336990935 p^{10} T^{8} - 22910390554828 p^{15} T^{9} + 1146261594 p^{20} T^{10} - 47500 p^{25} T^{11} + p^{30} T^{12} \)
59 \( 1 - 61744 T + 3765713234 T^{2} - 113008479343568 T^{3} + 3660429894448418007 T^{4} - \)\(64\!\cdots\!20\)\( T^{5} + \)\(20\!\cdots\!04\)\( T^{6} - \)\(64\!\cdots\!20\)\( p^{5} T^{7} + 3660429894448418007 p^{10} T^{8} - 113008479343568 p^{15} T^{9} + 3765713234 p^{20} T^{10} - 61744 p^{25} T^{11} + p^{30} T^{12} \)
61 \( 1 + 81581 T + 3670262887 T^{2} + 130450943818480 T^{3} + 3299287390789677309 T^{4} + \)\(71\!\cdots\!99\)\( T^{5} + \)\(19\!\cdots\!46\)\( T^{6} + \)\(71\!\cdots\!99\)\( p^{5} T^{7} + 3299287390789677309 p^{10} T^{8} + 130450943818480 p^{15} T^{9} + 3670262887 p^{20} T^{10} + 81581 p^{25} T^{11} + p^{30} T^{12} \)
67 \( 1 - 45756 T + 5521044082 T^{2} - 196315896858180 T^{3} + 13816249057775291495 T^{4} - \)\(39\!\cdots\!16\)\( T^{5} + \)\(22\!\cdots\!56\)\( T^{6} - \)\(39\!\cdots\!16\)\( p^{5} T^{7} + 13816249057775291495 p^{10} T^{8} - 196315896858180 p^{15} T^{9} + 5521044082 p^{20} T^{10} - 45756 p^{25} T^{11} + p^{30} T^{12} \)
71 \( 1 - 10416 T + 3015115818 T^{2} - 2714888754832 T^{3} + 4517274715949983647 T^{4} - \)\(86\!\cdots\!96\)\( T^{5} + \)\(95\!\cdots\!40\)\( T^{6} - \)\(86\!\cdots\!96\)\( p^{5} T^{7} + 4517274715949983647 p^{10} T^{8} - 2714888754832 p^{15} T^{9} + 3015115818 p^{20} T^{10} - 10416 p^{25} T^{11} + p^{30} T^{12} \)
73 \( 1 + 54615 T + 7587417371 T^{2} + 316656708585988 T^{3} + 27405357399303613849 T^{4} + \)\(14\!\cdots\!13\)\( p T^{5} + \)\(68\!\cdots\!78\)\( T^{6} + \)\(14\!\cdots\!13\)\( p^{6} T^{7} + 27405357399303613849 p^{10} T^{8} + 316656708585988 p^{15} T^{9} + 7587417371 p^{20} T^{10} + 54615 p^{25} T^{11} + p^{30} T^{12} \)
79 \( 1 - 145594 T + 25221486570 T^{2} - 2331155590535998 T^{3} + \)\(22\!\cdots\!51\)\( T^{4} - \)\(14\!\cdots\!84\)\( T^{5} + \)\(96\!\cdots\!44\)\( T^{6} - \)\(14\!\cdots\!84\)\( p^{5} T^{7} + \)\(22\!\cdots\!51\)\( p^{10} T^{8} - 2331155590535998 p^{15} T^{9} + 25221486570 p^{20} T^{10} - 145594 p^{25} T^{11} + p^{30} T^{12} \)
83 \( 1 - 160548 T + 24693624802 T^{2} - 2265316177470428 T^{3} + \)\(20\!\cdots\!99\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{5} + \)\(99\!\cdots\!32\)\( T^{6} - \)\(14\!\cdots\!12\)\( p^{5} T^{7} + \)\(20\!\cdots\!99\)\( p^{10} T^{8} - 2265316177470428 p^{15} T^{9} + 24693624802 p^{20} T^{10} - 160548 p^{25} T^{11} + p^{30} T^{12} \)
89 \( 1 + 97728 T + 15046266538 T^{2} + 906607292397504 T^{3} + 84370954871704719455 T^{4} + \)\(30\!\cdots\!56\)\( T^{5} + \)\(38\!\cdots\!96\)\( T^{6} + \)\(30\!\cdots\!56\)\( p^{5} T^{7} + 84370954871704719455 p^{10} T^{8} + 906607292397504 p^{15} T^{9} + 15046266538 p^{20} T^{10} + 97728 p^{25} T^{11} + p^{30} T^{12} \)
97 \( 1 + 760 T + 37269083226 T^{2} + 29277985984344 T^{3} + \)\(67\!\cdots\!87\)\( T^{4} + \)\(28\!\cdots\!96\)\( T^{5} + \)\(72\!\cdots\!32\)\( T^{6} + \)\(28\!\cdots\!96\)\( p^{5} T^{7} + \)\(67\!\cdots\!87\)\( p^{10} T^{8} + 29277985984344 p^{15} T^{9} + 37269083226 p^{20} T^{10} + 760 p^{25} T^{11} + p^{30} T^{12} \)
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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

−4.55983362871247484343840447147, −4.12567291737108379475055083564, −4.03477325268450676351575400754, −3.96489500501942111891036535060, −3.96032601502348594815120210532, −3.88902481471202629630378833329, −3.57034410730937716652565425372, −3.41347685772524094226001527494, −3.10198549590267424408913053670, −3.05608282014235226417626611135, −2.76340978845520624548583303013, −2.73457881778670053017560234210, −2.66703894874464617301262893119, −2.29417454733663311066615778472, −2.00464794572664972116128576236, −1.92594665863398284872737835602, −1.82854516711760839905729332069, −1.78774733948974790727167170097, −1.72004815440458286339452448643, −1.07224918201344201679586738608, −0.904569796265308037327283552690, −0.811007455588589758684490061343, −0.69042017713434991612757696343, −0.46291247749382633284912468113, −0.34230800808888997940606459344, 0.34230800808888997940606459344, 0.46291247749382633284912468113, 0.69042017713434991612757696343, 0.811007455588589758684490061343, 0.904569796265308037327283552690, 1.07224918201344201679586738608, 1.72004815440458286339452448643, 1.78774733948974790727167170097, 1.82854516711760839905729332069, 1.92594665863398284872737835602, 2.00464794572664972116128576236, 2.29417454733663311066615778472, 2.66703894874464617301262893119, 2.73457881778670053017560234210, 2.76340978845520624548583303013, 3.05608282014235226417626611135, 3.10198549590267424408913053670, 3.41347685772524094226001527494, 3.57034410730937716652565425372, 3.88902481471202629630378833329, 3.96032601502348594815120210532, 3.96489500501942111891036535060, 4.03477325268450676351575400754, 4.12567291737108379475055083564, 4.55983362871247484343840447147

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

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