Properties

Label 16-378e8-1.1-c3e8-0-4
Degree $16$
Conductor $4.168\times 10^{20}$
Sign $1$
Analytic cond. $6.12157\times 10^{10}$
Root an. cond. $4.72257$
Motivic weight $3$
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  + 8·2-s + 24·4-s − 5-s + 28·7-s − 8·10-s − 5·11-s + 21·13-s + 224·14-s − 240·16-s + 46·17-s − 188·19-s − 24·20-s − 40·22-s − 374·23-s + 230·25-s + 168·26-s + 672·28-s − 271·29-s + 243·31-s − 768·32-s + 368·34-s − 28·35-s − 362·37-s − 1.50e3·38-s + 213·41-s + 238·43-s − 120·44-s + ⋯
L(s)  = 1  + 2.82·2-s + 3·4-s − 0.0894·5-s + 1.51·7-s − 0.252·10-s − 0.137·11-s + 0.448·13-s + 4.27·14-s − 3.75·16-s + 0.656·17-s − 2.27·19-s − 0.268·20-s − 0.387·22-s − 3.39·23-s + 1.83·25-s + 1.26·26-s + 4.53·28-s − 1.73·29-s + 1.40·31-s − 4.24·32-s + 1.85·34-s − 0.135·35-s − 1.60·37-s − 6.42·38-s + 0.811·41-s + 0.844·43-s − 0.411·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{24} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(6.12157\times 10^{10}\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(24.14324022\)
\(L(\frac12)\) \(\approx\) \(24.14324022\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p T + p^{2} T^{2} )^{4} \)
3 \( 1 \)
7 \( ( 1 - p T + p^{2} T^{2} )^{4} \)
good5 \( 1 + T - 229 T^{2} - 774 p T^{3} + 26462 T^{4} + 678974 T^{5} + 1104902 p T^{6} - 2493309 p^{2} T^{7} - 997334831 T^{8} - 2493309 p^{5} T^{9} + 1104902 p^{7} T^{10} + 678974 p^{9} T^{11} + 26462 p^{12} T^{12} - 774 p^{16} T^{13} - 229 p^{18} T^{14} + p^{21} T^{15} + p^{24} T^{16} \)
11 \( 1 + 5 T - 2182 T^{2} - 158451 T^{3} + 2427827 T^{4} + 279627664 T^{5} + 9980471089 T^{6} - 295406959071 T^{7} - 17130347878430 T^{8} - 295406959071 p^{3} T^{9} + 9980471089 p^{6} T^{10} + 279627664 p^{9} T^{11} + 2427827 p^{12} T^{12} - 158451 p^{15} T^{13} - 2182 p^{18} T^{14} + 5 p^{21} T^{15} + p^{24} T^{16} \)
13 \( 1 - 21 T - 508 p T^{2} + 143907 T^{3} + 25185493 T^{4} - 450600480 T^{5} - 68065247446 T^{6} + 460246830006 T^{7} + 156900588822904 T^{8} + 460246830006 p^{3} T^{9} - 68065247446 p^{6} T^{10} - 450600480 p^{9} T^{11} + 25185493 p^{12} T^{12} + 143907 p^{15} T^{13} - 508 p^{19} T^{14} - 21 p^{21} T^{15} + p^{24} T^{16} \)
17 \( ( 1 - 23 T + 9893 T^{2} - 281573 T^{3} + 69040201 T^{4} - 281573 p^{3} T^{5} + 9893 p^{6} T^{6} - 23 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
19 \( ( 1 + 94 T + 13819 T^{2} + 1292485 T^{3} + 131846581 T^{4} + 1292485 p^{3} T^{5} + 13819 p^{6} T^{6} + 94 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
23 \( 1 + 374 T + 45833 T^{2} + 3344556 T^{3} + 859764083 T^{4} + 169835114038 T^{5} + 17527026209797 T^{6} + 1808539479332106 T^{7} + 222811406401501087 T^{8} + 1808539479332106 p^{3} T^{9} + 17527026209797 p^{6} T^{10} + 169835114038 p^{9} T^{11} + 859764083 p^{12} T^{12} + 3344556 p^{15} T^{13} + 45833 p^{18} T^{14} + 374 p^{21} T^{15} + p^{24} T^{16} \)
29 \( 1 + 271 T - 38803 T^{2} - 259344 p T^{3} + 3235076084 T^{4} + 303059385266 T^{5} - 105997401830390 T^{6} - 1727808215476197 T^{7} + 3339594708008059465 T^{8} - 1727808215476197 p^{3} T^{9} - 105997401830390 p^{6} T^{10} + 303059385266 p^{9} T^{11} + 3235076084 p^{12} T^{12} - 259344 p^{16} T^{13} - 38803 p^{18} T^{14} + 271 p^{21} T^{15} + p^{24} T^{16} \)
31 \( 1 - 243 T - 6151 T^{2} - 8914860 T^{3} + 2490318622 T^{4} + 117491450778 T^{5} + 42742085844890 T^{6} - 10211221082121813 T^{7} - 951217853496050879 T^{8} - 10211221082121813 p^{3} T^{9} + 42742085844890 p^{6} T^{10} + 117491450778 p^{9} T^{11} + 2490318622 p^{12} T^{12} - 8914860 p^{15} T^{13} - 6151 p^{18} T^{14} - 243 p^{21} T^{15} + p^{24} T^{16} \)
37 \( ( 1 + 181 T + 42844 T^{2} - 12321968 T^{3} - 3161946908 T^{4} - 12321968 p^{3} T^{5} + 42844 p^{6} T^{6} + 181 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
41 \( 1 - 213 T - 84590 T^{2} + 65999067 T^{3} - 3185697689 T^{4} - 5013788714586 T^{5} + 1428680211356995 T^{6} + 164868593391514353 T^{7} - \)\(12\!\cdots\!52\)\( T^{8} + 164868593391514353 p^{3} T^{9} + 1428680211356995 p^{6} T^{10} - 5013788714586 p^{9} T^{11} - 3185697689 p^{12} T^{12} + 65999067 p^{15} T^{13} - 84590 p^{18} T^{14} - 213 p^{21} T^{15} + p^{24} T^{16} \)
43 \( 1 - 238 T - 60675 T^{2} + 5271292 T^{3} - 2134721566 T^{4} + 2420831216454 T^{5} + 42843939153790 T^{6} - 184810086452079577 T^{7} + 50838540246961940772 T^{8} - 184810086452079577 p^{3} T^{9} + 42843939153790 p^{6} T^{10} + 2420831216454 p^{9} T^{11} - 2134721566 p^{12} T^{12} + 5271292 p^{15} T^{13} - 60675 p^{18} T^{14} - 238 p^{21} T^{15} + p^{24} T^{16} \)
47 \( 1 + 675 T + 37387 T^{2} - 77590386 T^{3} - 14073185960 T^{4} + 2780172589698 T^{5} + 544267648367926 T^{6} + 322605518680562283 T^{7} + \)\(21\!\cdots\!97\)\( T^{8} + 322605518680562283 p^{3} T^{9} + 544267648367926 p^{6} T^{10} + 2780172589698 p^{9} T^{11} - 14073185960 p^{12} T^{12} - 77590386 p^{15} T^{13} + 37387 p^{18} T^{14} + 675 p^{21} T^{15} + p^{24} T^{16} \)
53 \( ( 1 + 54 T + 236591 T^{2} + 19653129 T^{3} + 33755250168 T^{4} + 19653129 p^{3} T^{5} + 236591 p^{6} T^{6} + 54 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
59 \( 1 + 202 T - 531049 T^{2} - 16684212 T^{3} + 159300075830 T^{4} - 10988486623156 T^{5} - 38404349933253128 T^{6} + 1299442121783896557 T^{7} + \)\(80\!\cdots\!84\)\( T^{8} + 1299442121783896557 p^{3} T^{9} - 38404349933253128 p^{6} T^{10} - 10988486623156 p^{9} T^{11} + 159300075830 p^{12} T^{12} - 16684212 p^{15} T^{13} - 531049 p^{18} T^{14} + 202 p^{21} T^{15} + p^{24} T^{16} \)
61 \( 1 - 1212 T + 239267 T^{2} - 3162822 T^{3} + 187949681281 T^{4} - 58043170902294 T^{5} - 44955529435193749 T^{6} + 9664786778799768882 T^{7} + \)\(71\!\cdots\!33\)\( T^{8} + 9664786778799768882 p^{3} T^{9} - 44955529435193749 p^{6} T^{10} - 58043170902294 p^{9} T^{11} + 187949681281 p^{12} T^{12} - 3162822 p^{15} T^{13} + 239267 p^{18} T^{14} - 1212 p^{21} T^{15} + p^{24} T^{16} \)
67 \( 1 + 139 T - 800538 T^{2} - 280034389 T^{3} + 334951438031 T^{4} + 134625567082980 T^{5} - 75122243735690903 T^{6} - 22025505408916196753 T^{7} + \)\(16\!\cdots\!74\)\( T^{8} - 22025505408916196753 p^{3} T^{9} - 75122243735690903 p^{6} T^{10} + 134625567082980 p^{9} T^{11} + 334951438031 p^{12} T^{12} - 280034389 p^{15} T^{13} - 800538 p^{18} T^{14} + 139 p^{21} T^{15} + p^{24} T^{16} \)
71 \( ( 1 - 1295 T + 1795244 T^{2} - 1349389658 T^{3} + 1011484858822 T^{4} - 1349389658 p^{3} T^{5} + 1795244 p^{6} T^{6} - 1295 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
73 \( ( 1 + 2000 T + 2663509 T^{2} + 2429329973 T^{3} + 1727799097657 T^{4} + 2429329973 p^{3} T^{5} + 2663509 p^{6} T^{6} + 2000 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
79 \( 1 - 1545 T + 135641 T^{2} + 576786432 T^{3} + 262453195432 T^{4} - 392965726353228 T^{5} - 106866556646468452 T^{6} + 77122026263059272753 T^{7} + \)\(55\!\cdots\!45\)\( T^{8} + 77122026263059272753 p^{3} T^{9} - 106866556646468452 p^{6} T^{10} - 392965726353228 p^{9} T^{11} + 262453195432 p^{12} T^{12} + 576786432 p^{15} T^{13} + 135641 p^{18} T^{14} - 1545 p^{21} T^{15} + p^{24} T^{16} \)
83 \( 1 - 142 T - 981853 T^{2} - 1435324800 T^{3} + 772358229251 T^{4} + 1109364392322268 T^{5} + 9657132248274443 p T^{6} - \)\(60\!\cdots\!94\)\( T^{7} - \)\(56\!\cdots\!45\)\( T^{8} - \)\(60\!\cdots\!94\)\( p^{3} T^{9} + 9657132248274443 p^{7} T^{10} + 1109364392322268 p^{9} T^{11} + 772358229251 p^{12} T^{12} - 1435324800 p^{15} T^{13} - 981853 p^{18} T^{14} - 142 p^{21} T^{15} + p^{24} T^{16} \)
89 \( ( 1 - 132 T + 738863 T^{2} + 1004820993 T^{3} - 79516549278 T^{4} + 1004820993 p^{3} T^{5} + 738863 p^{6} T^{6} - 132 p^{9} T^{7} + p^{12} T^{8} )^{2} \)
97 \( 1 - 638 T - 197967 T^{2} + 1627259216 T^{3} - 1785731534086 T^{4} + 398431063574808 T^{5} + 521780813880337198 T^{6} - \)\(12\!\cdots\!03\)\( T^{7} + \)\(97\!\cdots\!70\)\( T^{8} - \)\(12\!\cdots\!03\)\( p^{3} T^{9} + 521780813880337198 p^{6} T^{10} + 398431063574808 p^{9} T^{11} - 1785731534086 p^{12} T^{12} + 1627259216 p^{15} T^{13} - 197967 p^{18} T^{14} - 638 p^{21} T^{15} + p^{24} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.51271095943323720170157622103, −4.39157467760278231066282747268, −4.36736047393795921227973070883, −4.21480469841924648870094383713, −4.21173254963069663839274743612, −3.75498282317031553852661752354, −3.64629548299120769877388007732, −3.60122729059819026403518602558, −3.37702222496542764486036966361, −3.34411234360318503053717577858, −3.29291408924588116341535042667, −2.97264291917788316360169964469, −2.70600873642226499598359810763, −2.41718474876346141865011995587, −2.34307088485359895080241297216, −2.24005255182810362886395252302, −2.02147666517202729179636933391, −1.86740887697764183549218804193, −1.75524533632562382693006999037, −1.39535468258558520289664800205, −1.14440433553776294391225634319, −1.00556095825182094121486418953, −0.45689479608477871996306945224, −0.34164719674664552765800784340, −0.32783636164795630891351928297, 0.32783636164795630891351928297, 0.34164719674664552765800784340, 0.45689479608477871996306945224, 1.00556095825182094121486418953, 1.14440433553776294391225634319, 1.39535468258558520289664800205, 1.75524533632562382693006999037, 1.86740887697764183549218804193, 2.02147666517202729179636933391, 2.24005255182810362886395252302, 2.34307088485359895080241297216, 2.41718474876346141865011995587, 2.70600873642226499598359810763, 2.97264291917788316360169964469, 3.29291408924588116341535042667, 3.34411234360318503053717577858, 3.37702222496542764486036966361, 3.60122729059819026403518602558, 3.64629548299120769877388007732, 3.75498282317031553852661752354, 4.21173254963069663839274743612, 4.21480469841924648870094383713, 4.36736047393795921227973070883, 4.39157467760278231066282747268, 4.51271095943323720170157622103

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

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