Dirichlet series
L(s) = 1 | + 1.28e6·4-s + 1.47e5·5-s + 4.82e9·9-s − 3.37e9·11-s + 5.37e11·16-s + 4.54e12·19-s + 1.89e11·20-s + 8.86e12·25-s − 1.88e14·29-s + 7.21e13·31-s + 6.20e15·36-s − 3.31e15·41-s − 4.34e15·44-s + 7.08e14·45-s + 4.00e16·49-s − 4.96e14·55-s + 2.59e17·59-s − 5.82e17·61-s + 4.13e16·64-s − 2.61e18·71-s + 5.85e18·76-s + 3.93e18·79-s + 7.89e16·80-s + 9.64e18·81-s + 2.12e18·89-s + 6.68e17·95-s − 1.62e19·99-s + ⋯ |
L(s) = 1 | + 2.45·4-s + 0.0336·5-s + 4.14·9-s − 0.432·11-s + 1.95·16-s + 3.23·19-s + 0.0826·20-s + 0.464·25-s − 2.40·29-s + 0.489·31-s + 10.1·36-s − 1.58·41-s − 1.06·44-s + 0.139·45-s + 3.51·49-s − 0.0145·55-s + 3.90·59-s − 6.38·61-s + 0.287·64-s − 6.77·71-s + 7.93·76-s + 3.69·79-s + 0.0657·80-s + 7.13·81-s + 0.641·89-s + 0.108·95-s − 1.79·99-s + ⋯ |
Functional equation
Invariants
Degree: | \(16\) |
Conductor: | \(390625\) = \(5^{8}\) |
Sign: | $1$ |
Analytic conductor: | \(2.93536\times 10^{8}\) |
Root analytic conductor: | \(3.38243\) |
Motivic weight: | \(19\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((16,\ 390625,\ (\ :[19/2]^{8}),\ 1)\) |
Particular Values
\(L(10)\) | \(\approx\) | \(31.13554850\) |
\(L(\frac12)\) | \(\approx\) | \(31.13554850\) |
\(L(\frac{21}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 5 | \( 1 - 1176 p^{3} T - 113242244 p^{7} T^{2} + 15081850008 p^{12} T^{3} + 389129703582 p^{21} T^{4} + 15081850008 p^{31} T^{5} - 113242244 p^{45} T^{6} - 1176 p^{60} T^{7} + p^{76} T^{8} \) |
good | 2 | \( 1 - 321695 p^{2} T^{2} + 4369211371 p^{8} T^{4} - 12043934255735 p^{16} T^{6} + 6568673708573269 p^{26} T^{8} - 12043934255735 p^{54} T^{10} + 4369211371 p^{84} T^{12} - 321695 p^{116} T^{14} + p^{152} T^{16} \) |
3 | \( 1 - 535747240 p^{2} T^{2} + 18664213771033564 p^{6} T^{4} - \)\(15\!\cdots\!40\)\( p^{13} T^{6} + \)\(89\!\cdots\!34\)\( p^{18} T^{8} - \)\(15\!\cdots\!40\)\( p^{51} T^{10} + 18664213771033564 p^{82} T^{12} - 535747240 p^{116} T^{14} + p^{152} T^{16} \) | |
7 | \( 1 - 40030951735584200 T^{2} + \)\(23\!\cdots\!72\)\( p^{3} T^{4} - \)\(47\!\cdots\!00\)\( p^{4} T^{6} + \)\(24\!\cdots\!06\)\( p^{8} T^{8} - \)\(47\!\cdots\!00\)\( p^{42} T^{10} + \)\(23\!\cdots\!72\)\( p^{79} T^{12} - 40030951735584200 p^{114} T^{14} + p^{152} T^{16} \) | |
11 | \( ( 1 + 1689787632 T + 44995983407219867948 T^{2} + \)\(51\!\cdots\!44\)\( p T^{3} + \)\(12\!\cdots\!70\)\( p^{2} T^{4} + \)\(51\!\cdots\!44\)\( p^{20} T^{5} + 44995983407219867948 p^{38} T^{6} + 1689787632 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
13 | \( 1 - \)\(73\!\cdots\!20\)\( T^{2} + \)\(16\!\cdots\!64\)\( p^{2} T^{4} - \)\(23\!\cdots\!40\)\( p^{4} T^{6} + \)\(24\!\cdots\!94\)\( p^{6} T^{8} - \)\(23\!\cdots\!40\)\( p^{42} T^{10} + \)\(16\!\cdots\!64\)\( p^{78} T^{12} - \)\(73\!\cdots\!20\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
17 | \( 1 - \)\(52\!\cdots\!60\)\( p^{2} T^{2} + \)\(72\!\cdots\!48\)\( p^{5} T^{4} - \)\(10\!\cdots\!60\)\( p^{7} T^{6} + \)\(17\!\cdots\!46\)\( p^{8} T^{8} - \)\(10\!\cdots\!60\)\( p^{45} T^{10} + \)\(72\!\cdots\!48\)\( p^{81} T^{12} - \)\(52\!\cdots\!60\)\( p^{116} T^{14} + p^{152} T^{16} \) | |
19 | \( ( 1 - 2273594190320 T + \)\(24\!\cdots\!64\)\( p T^{2} - \)\(77\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!46\)\( T^{4} - \)\(77\!\cdots\!40\)\( p^{19} T^{5} + \)\(24\!\cdots\!64\)\( p^{39} T^{6} - 2273594190320 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
23 | \( 1 - \)\(26\!\cdots\!80\)\( T^{2} + \)\(48\!\cdots\!76\)\( T^{4} - \)\(55\!\cdots\!60\)\( T^{6} + \)\(48\!\cdots\!66\)\( T^{8} - \)\(55\!\cdots\!60\)\( p^{38} T^{10} + \)\(48\!\cdots\!76\)\( p^{76} T^{12} - \)\(26\!\cdots\!80\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
29 | \( ( 1 + 94111150172520 T + \)\(18\!\cdots\!76\)\( T^{2} + \)\(85\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} + \)\(85\!\cdots\!40\)\( p^{19} T^{5} + \)\(18\!\cdots\!76\)\( p^{38} T^{6} + 94111150172520 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
31 | \( ( 1 - 36057503343488 T + \)\(32\!\cdots\!88\)\( T^{2} - \)\(22\!\cdots\!36\)\( T^{3} + \)\(45\!\cdots\!70\)\( T^{4} - \)\(22\!\cdots\!36\)\( p^{19} T^{5} + \)\(32\!\cdots\!88\)\( p^{38} T^{6} - 36057503343488 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
37 | \( 1 - \)\(22\!\cdots\!20\)\( T^{2} + \)\(21\!\cdots\!16\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{6} + \)\(72\!\cdots\!46\)\( T^{8} - \)\(13\!\cdots\!40\)\( p^{38} T^{10} + \)\(21\!\cdots\!16\)\( p^{76} T^{12} - \)\(22\!\cdots\!20\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
41 | \( ( 1 + 1657134714428952 T + \)\(15\!\cdots\!08\)\( T^{2} + \)\(20\!\cdots\!04\)\( T^{3} + \)\(95\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!04\)\( p^{19} T^{5} + \)\(15\!\cdots\!08\)\( p^{38} T^{6} + 1657134714428952 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
43 | \( 1 - \)\(66\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!96\)\( T^{4} - \)\(39\!\cdots\!00\)\( T^{6} + \)\(52\!\cdots\!06\)\( T^{8} - \)\(39\!\cdots\!00\)\( p^{38} T^{10} + \)\(20\!\cdots\!96\)\( p^{76} T^{12} - \)\(66\!\cdots\!00\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
47 | \( 1 - \)\(24\!\cdots\!60\)\( T^{2} + \)\(31\!\cdots\!56\)\( T^{4} - \)\(28\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!26\)\( T^{8} - \)\(28\!\cdots\!20\)\( p^{38} T^{10} + \)\(31\!\cdots\!56\)\( p^{76} T^{12} - \)\(24\!\cdots\!60\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
53 | \( 1 - \)\(19\!\cdots\!60\)\( T^{2} + \)\(24\!\cdots\!56\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!26\)\( T^{8} - \)\(20\!\cdots\!20\)\( p^{38} T^{10} + \)\(24\!\cdots\!56\)\( p^{76} T^{12} - \)\(19\!\cdots\!60\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
59 | \( ( 1 - 129824016389006160 T + \)\(17\!\cdots\!56\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!26\)\( T^{4} - \)\(14\!\cdots\!20\)\( p^{19} T^{5} + \)\(17\!\cdots\!56\)\( p^{38} T^{6} - 129824016389006160 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
61 | \( ( 1 + 291446781376613032 T + \)\(58\!\cdots\!48\)\( T^{2} + \)\(77\!\cdots\!84\)\( T^{3} + \)\(82\!\cdots\!70\)\( T^{4} + \)\(77\!\cdots\!84\)\( p^{19} T^{5} + \)\(58\!\cdots\!48\)\( p^{38} T^{6} + 291446781376613032 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
67 | \( 1 - \)\(24\!\cdots\!40\)\( T^{2} + \)\(29\!\cdots\!36\)\( T^{4} - \)\(23\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!86\)\( T^{8} - \)\(23\!\cdots\!80\)\( p^{38} T^{10} + \)\(29\!\cdots\!36\)\( p^{76} T^{12} - \)\(24\!\cdots\!40\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
71 | \( ( 1 + 1308503800301463072 T + \)\(93\!\cdots\!68\)\( T^{2} + \)\(48\!\cdots\!24\)\( T^{3} + \)\(20\!\cdots\!70\)\( T^{4} + \)\(48\!\cdots\!24\)\( p^{19} T^{5} + \)\(93\!\cdots\!68\)\( p^{38} T^{6} + 1308503800301463072 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
73 | \( 1 - \)\(77\!\cdots\!80\)\( T^{2} + \)\(33\!\cdots\!76\)\( T^{4} - \)\(95\!\cdots\!60\)\( T^{6} + \)\(24\!\cdots\!66\)\( T^{8} - \)\(95\!\cdots\!60\)\( p^{38} T^{10} + \)\(33\!\cdots\!76\)\( p^{76} T^{12} - \)\(77\!\cdots\!80\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
79 | \( ( 1 - 1967546045554930880 T + \)\(57\!\cdots\!76\)\( T^{2} - \)\(69\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!66\)\( T^{4} - \)\(69\!\cdots\!60\)\( p^{19} T^{5} + \)\(57\!\cdots\!76\)\( p^{38} T^{6} - 1967546045554930880 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
83 | \( 1 - \)\(12\!\cdots\!40\)\( T^{2} + \)\(74\!\cdots\!36\)\( T^{4} - \)\(30\!\cdots\!80\)\( T^{6} + \)\(99\!\cdots\!86\)\( T^{8} - \)\(30\!\cdots\!80\)\( p^{38} T^{10} + \)\(74\!\cdots\!36\)\( p^{76} T^{12} - \)\(12\!\cdots\!40\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
89 | \( ( 1 - 1060070550818460840 T + \)\(28\!\cdots\!36\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(40\!\cdots\!86\)\( T^{4} - \)\(12\!\cdots\!80\)\( p^{19} T^{5} + \)\(28\!\cdots\!36\)\( p^{38} T^{6} - 1060070550818460840 p^{57} T^{7} + p^{76} T^{8} )^{2} \) | |
97 | \( 1 - \)\(32\!\cdots\!60\)\( T^{2} + \)\(51\!\cdots\!56\)\( T^{4} - \)\(50\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!26\)\( T^{8} - \)\(50\!\cdots\!20\)\( p^{38} T^{10} + \)\(51\!\cdots\!56\)\( p^{76} T^{12} - \)\(32\!\cdots\!60\)\( p^{114} T^{14} + p^{152} T^{16} \) | |
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Imaginary part of the first few zeros on the critical line
−7.39573164421913238834310301911, −7.25883133510291488012789057423, −7.04473472253404703331352963041, −6.61734897906604316339147424740, −6.33679891528075839509365594569, −6.29600916637654116772491843110, −5.57642900716856323219850371192, −5.45240308834153565353557527992, −5.31572913527155533852192511861, −4.75903219157888522097274595731, −4.35891039511971729915563440192, −4.32148374013859165094166361759, −3.98130355863727855174632213388, −3.62906517344969903661762936904, −3.07478043977055208750766695212, −3.01561736054509416054805534549, −2.82873058678629568874298440152, −2.21183286693286771553106214346, −1.80674809275836228471380955443, −1.74882732092898545758802220120, −1.69129196686179634757714001565, −1.31084057141007667986305372199, −0.920789572771209375375736521027, −0.78821774480713855965924585185, −0.31970905131449460091552508910, 0.31970905131449460091552508910, 0.78821774480713855965924585185, 0.920789572771209375375736521027, 1.31084057141007667986305372199, 1.69129196686179634757714001565, 1.74882732092898545758802220120, 1.80674809275836228471380955443, 2.21183286693286771553106214346, 2.82873058678629568874298440152, 3.01561736054509416054805534549, 3.07478043977055208750766695212, 3.62906517344969903661762936904, 3.98130355863727855174632213388, 4.32148374013859165094166361759, 4.35891039511971729915563440192, 4.75903219157888522097274595731, 5.31572913527155533852192511861, 5.45240308834153565353557527992, 5.57642900716856323219850371192, 6.29600916637654116772491843110, 6.33679891528075839509365594569, 6.61734897906604316339147424740, 7.04473472253404703331352963041, 7.25883133510291488012789057423, 7.39573164421913238834310301911