Properties

Degree $16$
Conductor $4.348\times 10^{12}$
Sign $1$
Motivic weight $4$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·4-s + 18·5-s − 162·7-s + 190·9-s − 6·11-s + 640·16-s + 510·17-s − 12·19-s − 576·20-s − 396·23-s − 609·25-s + 5.18e3·28-s − 2.91e3·35-s − 6.08e3·36-s − 8.65e3·43-s + 192·44-s + 3.42e3·45-s + 3.21e3·47-s + 8.12e3·49-s − 108·55-s + 1.31e3·61-s − 3.07e4·63-s − 1.02e4·64-s − 1.63e4·68-s + 2.33e4·73-s + 384·76-s + 972·77-s + ⋯
L(s)  = 1  − 2·4-s + 0.719·5-s − 3.30·7-s + 2.34·9-s − 0.0495·11-s + 5/2·16-s + 1.76·17-s − 0.0332·19-s − 1.43·20-s − 0.748·23-s − 0.974·25-s + 6.61·28-s − 2.38·35-s − 4.69·36-s − 4.68·43-s + 0.0991·44-s + 1.68·45-s + 1.45·47-s + 3.38·49-s − 0.0357·55-s + 0.353·61-s − 7.75·63-s − 5/2·64-s − 3.52·68-s + 4.39·73-s + 0.0664·76-s + 0.163·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 19^{8}\)
Sign: $1$
Motivic weight: \(4\)
Character: induced by $\chi_{38} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 19^{8} ,\ ( \ : [2]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.02036\)
\(L(\frac12)\) \(\approx\) \(1.02036\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{3} T^{2} )^{4} \)
19 \( 1 + 12 T + 303560 T^{2} - 1063956 p T^{3} + 6709482 p^{3} T^{4} - 1063956 p^{5} T^{5} + 303560 p^{8} T^{6} + 12 p^{12} T^{7} + p^{16} T^{8} \)
good3 \( 1 - 190 T^{2} + 9523 p T^{4} - 297718 p^{2} T^{6} + 114172 p^{7} T^{8} - 297718 p^{10} T^{10} + 9523 p^{17} T^{12} - 190 p^{24} T^{14} + p^{32} T^{16} \)
5 \( ( 1 - 9 T + 426 T^{2} - 10251 T^{3} + 716146 T^{4} - 10251 p^{4} T^{5} + 426 p^{8} T^{6} - 9 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
7 \( ( 1 + 81 T + 5777 T^{2} + 49050 p T^{3} + 20657670 T^{4} + 49050 p^{5} T^{5} + 5777 p^{8} T^{6} + 81 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
11 \( ( 1 + 3 T + 20874 T^{2} + 203751 p T^{3} + 23528702 p T^{4} + 203751 p^{5} T^{5} + 20874 p^{8} T^{6} + 3 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
13 \( 1 - 87902 T^{2} + 4031399833 T^{4} - 153863832997766 T^{6} + 5034444913352767444 T^{8} - 153863832997766 p^{8} T^{10} + 4031399833 p^{16} T^{12} - 87902 p^{24} T^{14} + p^{32} T^{16} \)
17 \( ( 1 - 15 p T + 266529 T^{2} - 48908322 T^{3} + 30973446334 T^{4} - 48908322 p^{4} T^{5} + 266529 p^{8} T^{6} - 15 p^{13} T^{7} + p^{16} T^{8} )^{2} \)
23 \( ( 1 + 198 T + 793305 T^{2} + 151845534 T^{3} + 305726793412 T^{4} + 151845534 p^{4} T^{5} + 793305 p^{8} T^{6} + 198 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
29 \( 1 - 1884206 T^{2} + 2536130733097 T^{4} - 2411241746303389430 T^{6} + \)\(19\!\cdots\!76\)\( T^{8} - 2411241746303389430 p^{8} T^{10} + 2536130733097 p^{16} T^{12} - 1884206 p^{24} T^{14} + p^{32} T^{16} \)
31 \( 1 - 5185760 T^{2} + 13147692978364 T^{4} - 21074811809060168480 T^{6} + \)\(23\!\cdots\!86\)\( T^{8} - 21074811809060168480 p^{8} T^{10} + 13147692978364 p^{16} T^{12} - 5185760 p^{24} T^{14} + p^{32} T^{16} \)
37 \( 1 - 8989760 T^{2} + 32058963480700 T^{4} - 61405971768293818304 T^{6} + \)\(98\!\cdots\!62\)\( T^{8} - 61405971768293818304 p^{8} T^{10} + 32058963480700 p^{16} T^{12} - 8989760 p^{24} T^{14} + p^{32} T^{16} \)
41 \( 1 - 15418400 T^{2} + 116131174644796 T^{4} - \)\(56\!\cdots\!20\)\( T^{6} + \)\(18\!\cdots\!30\)\( T^{8} - \)\(56\!\cdots\!20\)\( p^{8} T^{10} + 116131174644796 p^{16} T^{12} - 15418400 p^{24} T^{14} + p^{32} T^{16} \)
43 \( ( 1 + 4327 T + 17040418 T^{2} + 41562268261 T^{3} + 93546947175394 T^{4} + 41562268261 p^{4} T^{5} + 17040418 p^{8} T^{6} + 4327 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
47 \( ( 1 - 1605 T + 15846978 T^{2} - 24977693547 T^{3} + 106935047396698 T^{4} - 24977693547 p^{4} T^{5} + 15846978 p^{8} T^{6} - 1605 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
53 \( 1 - 26276750 T^{2} + 354841909979401 T^{4} - \)\(39\!\cdots\!70\)\( T^{6} + \)\(37\!\cdots\!60\)\( T^{8} - \)\(39\!\cdots\!70\)\( p^{8} T^{10} + 354841909979401 p^{16} T^{12} - 26276750 p^{24} T^{14} + p^{32} T^{16} \)
59 \( 1 - 70986542 T^{2} + 2266647860250313 T^{4} - \)\(44\!\cdots\!70\)\( T^{6} + \)\(62\!\cdots\!16\)\( T^{8} - \)\(44\!\cdots\!70\)\( p^{8} T^{10} + 2266647860250313 p^{16} T^{12} - 70986542 p^{24} T^{14} + p^{32} T^{16} \)
61 \( ( 1 - 657 T + 13651298 T^{2} + 36749564541 T^{3} + 190575679355970 T^{4} + 36749564541 p^{4} T^{5} + 13651298 p^{8} T^{6} - 657 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
67 \( 1 - 11587598 T^{2} + 1243984422721705 T^{4} - \)\(11\!\cdots\!74\)\( T^{6} + \)\(71\!\cdots\!88\)\( T^{8} - \)\(11\!\cdots\!74\)\( p^{8} T^{10} + 1243984422721705 p^{16} T^{12} - 11587598 p^{24} T^{14} + p^{32} T^{16} \)
71 \( 1 - 27489248 T^{2} + 2078050971545020 T^{4} - \)\(45\!\cdots\!44\)\( T^{6} + \)\(19\!\cdots\!38\)\( T^{8} - \)\(45\!\cdots\!44\)\( p^{8} T^{10} + 2078050971545020 p^{16} T^{12} - 27489248 p^{24} T^{14} + p^{32} T^{16} \)
73 \( ( 1 - 11699 T + 88585093 T^{2} - 366782768102 T^{3} + 1847388837013414 T^{4} - 366782768102 p^{4} T^{5} + 88585093 p^{8} T^{6} - 11699 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
79 \( 1 - 198341024 T^{2} + 17865606744931900 T^{4} - \)\(10\!\cdots\!64\)\( T^{6} + \)\(43\!\cdots\!70\)\( T^{8} - \)\(10\!\cdots\!64\)\( p^{8} T^{10} + 17865606744931900 p^{16} T^{12} - 198341024 p^{24} T^{14} + p^{32} T^{16} \)
83 \( ( 1 + 5220 T + 132074424 T^{2} + 739113578028 T^{3} + 8089121462686894 T^{4} + 739113578028 p^{4} T^{5} + 132074424 p^{8} T^{6} + 5220 p^{12} T^{7} + p^{16} T^{8} )^{2} \)
89 \( 1 - 219510728 T^{2} + 29593428331358620 T^{4} - \)\(27\!\cdots\!44\)\( T^{6} + \)\(20\!\cdots\!18\)\( T^{8} - \)\(27\!\cdots\!44\)\( p^{8} T^{10} + 29593428331358620 p^{16} T^{12} - 219510728 p^{24} T^{14} + p^{32} T^{16} \)
97 \( 1 - 272222816 T^{2} + 39430845297013180 T^{4} - \)\(47\!\cdots\!04\)\( T^{6} + \)\(48\!\cdots\!14\)\( T^{8} - \)\(47\!\cdots\!04\)\( p^{8} T^{10} + 39430845297013180 p^{16} T^{12} - 272222816 p^{24} T^{14} + p^{32} T^{16} \)
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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

−7.11650355526117091921472838055, −6.82087467934725025219984456751, −6.75383165481682875017147314444, −6.40372248941917225152295842607, −6.33767453472833263068159137589, −6.17309335710984374026919559034, −5.97416401507610809860735011924, −5.67503612724174396467737138668, −5.38000098369012329067096369423, −5.07252253047712024793266339879, −5.04864726349963879376606642930, −4.90278258189868764941967809154, −4.34314650538667650447278291589, −4.22969103123084269542977091926, −3.89393897250238198753432320968, −3.58477140934765143233924250986, −3.51547206206605362420800676381, −3.45850268069678081648832984730, −2.99676737039772214886529090026, −2.66338611845239073724008034363, −1.91320333518087895849646334599, −1.71865712054477534848608842638, −1.22290137124919347744828337817, −0.56051810683536469085704849293, −0.31947384406043978224097145339, 0.31947384406043978224097145339, 0.56051810683536469085704849293, 1.22290137124919347744828337817, 1.71865712054477534848608842638, 1.91320333518087895849646334599, 2.66338611845239073724008034363, 2.99676737039772214886529090026, 3.45850268069678081648832984730, 3.51547206206605362420800676381, 3.58477140934765143233924250986, 3.89393897250238198753432320968, 4.22969103123084269542977091926, 4.34314650538667650447278291589, 4.90278258189868764941967809154, 5.04864726349963879376606642930, 5.07252253047712024793266339879, 5.38000098369012329067096369423, 5.67503612724174396467737138668, 5.97416401507610809860735011924, 6.17309335710984374026919559034, 6.33767453472833263068159137589, 6.40372248941917225152295842607, 6.75383165481682875017147314444, 6.82087467934725025219984456751, 7.11650355526117091921472838055

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

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