Properties

Label 16-162e8-1.1-c7e8-0-0
Degree $16$
Conductor $4.744\times 10^{17}$
Sign $1$
Analytic cond. $4.30171\times 10^{13}$
Root an. cond. $7.11381$
Motivic weight $7$
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  + 32·2-s + 384·4-s + 528·5-s − 560·7-s + 1.68e4·10-s + 2.16e3·11-s − 1.34e4·13-s − 1.79e4·14-s − 6.14e4·16-s − 4.51e4·17-s + 7.34e4·19-s + 2.02e5·20-s + 6.91e4·22-s + 6.26e4·23-s + 2.48e5·25-s − 4.30e5·26-s − 2.15e5·28-s + 6.84e4·29-s − 2.27e5·31-s − 7.86e5·32-s − 1.44e6·34-s − 2.95e5·35-s + 1.04e6·37-s + 2.34e6·38-s + 6.72e4·41-s − 5.62e5·43-s + 8.29e5·44-s + ⋯
L(s)  = 1  + 2.82·2-s + 3·4-s + 1.88·5-s − 0.617·7-s + 5.34·10-s + 0.489·11-s − 1.69·13-s − 1.74·14-s − 3.75·16-s − 2.22·17-s + 2.45·19-s + 5.66·20-s + 1.38·22-s + 1.07·23-s + 3.17·25-s − 4.80·26-s − 1.85·28-s + 0.520·29-s − 1.37·31-s − 4.24·32-s − 6.30·34-s − 1.16·35-s + 3.39·37-s + 6.94·38-s + 0.152·41-s − 1.07·43-s + 1.46·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{32}\)
Sign: $1$
Analytic conductor: \(4.30171\times 10^{13}\)
Root analytic conductor: \(7.11381\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [7/2]^{8} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(35.79142929\)
\(L(\frac12)\) \(\approx\) \(35.79142929\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p^{3} T + p^{6} T^{2} )^{4} \)
3 \( 1 \)
good5 \( 1 - 528 T + 6098 p T^{2} + 28717728 T^{3} - 6646473479 T^{4} + 296584780896 p T^{5} - 2892551971246 p^{3} T^{6} - 302393999457264 p^{3} T^{7} + 55894546867697956 p^{4} T^{8} - 302393999457264 p^{10} T^{9} - 2892551971246 p^{17} T^{10} + 296584780896 p^{22} T^{11} - 6646473479 p^{28} T^{12} + 28717728 p^{35} T^{13} + 6098 p^{43} T^{14} - 528 p^{49} T^{15} + p^{56} T^{16} \)
7 \( 1 + 80 p T - 765588 T^{2} + 4467040 p^{2} T^{3} + 383373171338 T^{4} - 7906783464240 p^{2} T^{5} + 685455739329691696 T^{6} + 72837084001705034960 p T^{7} - \)\(47\!\cdots\!21\)\( T^{8} + 72837084001705034960 p^{8} T^{9} + 685455739329691696 p^{14} T^{10} - 7906783464240 p^{23} T^{11} + 383373171338 p^{28} T^{12} + 4467040 p^{37} T^{13} - 765588 p^{42} T^{14} + 80 p^{50} T^{15} + p^{56} T^{16} \)
11 \( 1 - 2160 T - 34124660 T^{2} + 31418508960 T^{3} + 360882118725466 T^{4} + 1046783223117817200 T^{5} - \)\(91\!\cdots\!20\)\( T^{6} - \)\(19\!\cdots\!80\)\( T^{7} + \)\(32\!\cdots\!75\)\( T^{8} - \)\(19\!\cdots\!80\)\( p^{7} T^{9} - \)\(91\!\cdots\!20\)\( p^{14} T^{10} + 1046783223117817200 p^{21} T^{11} + 360882118725466 p^{28} T^{12} + 31418508960 p^{35} T^{13} - 34124660 p^{42} T^{14} - 2160 p^{49} T^{15} + p^{56} T^{16} \)
13 \( 1 + 13460 T - 2381874 T^{2} - 1006670971640 T^{3} - 486455445387667 p T^{4} - 19004114515218025800 T^{5} - \)\(49\!\cdots\!06\)\( T^{6} + \)\(18\!\cdots\!80\)\( T^{7} + \)\(27\!\cdots\!24\)\( T^{8} + \)\(18\!\cdots\!80\)\( p^{7} T^{9} - \)\(49\!\cdots\!06\)\( p^{14} T^{10} - 19004114515218025800 p^{21} T^{11} - 486455445387667 p^{29} T^{12} - 1006670971640 p^{35} T^{13} - 2381874 p^{42} T^{14} + 13460 p^{49} T^{15} + p^{56} T^{16} \)
17 \( ( 1 + 22560 T + 652500158 T^{2} + 256565445120 T^{3} + 29387435907228099 T^{4} + 256565445120 p^{7} T^{5} + 652500158 p^{14} T^{6} + 22560 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
19 \( ( 1 - 36704 T + 3047986756 T^{2} - 91971206881760 T^{3} + 3917212049965000630 T^{4} - 91971206881760 p^{7} T^{5} + 3047986756 p^{14} T^{6} - 36704 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
23 \( 1 - 62640 T - 285060028 p T^{2} + 64514052819360 T^{3} + 43613736591780304234 T^{4} + \)\(46\!\cdots\!00\)\( T^{5} - \)\(16\!\cdots\!96\)\( T^{6} - \)\(37\!\cdots\!40\)\( p T^{7} + \)\(68\!\cdots\!31\)\( p^{2} T^{8} - \)\(37\!\cdots\!40\)\( p^{8} T^{9} - \)\(16\!\cdots\!96\)\( p^{14} T^{10} + \)\(46\!\cdots\!00\)\( p^{21} T^{11} + 43613736591780304234 p^{28} T^{12} + 64514052819360 p^{35} T^{13} - 285060028 p^{43} T^{14} - 62640 p^{49} T^{15} + p^{56} T^{16} \)
29 \( 1 - 68400 T - 22912702982 T^{2} - 3119287342989600 T^{3} + \)\(41\!\cdots\!29\)\( T^{4} + \)\(81\!\cdots\!00\)\( T^{5} + \)\(67\!\cdots\!94\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} - \)\(12\!\cdots\!80\)\( T^{8} - \)\(12\!\cdots\!00\)\( p^{7} T^{9} + \)\(67\!\cdots\!94\)\( p^{14} T^{10} + \)\(81\!\cdots\!00\)\( p^{21} T^{11} + \)\(41\!\cdots\!29\)\( p^{28} T^{12} - 3119287342989600 p^{35} T^{13} - 22912702982 p^{42} T^{14} - 68400 p^{49} T^{15} + p^{56} T^{16} \)
31 \( 1 + 227504 T - 21465016380 T^{2} + 4801891406924128 T^{3} + \)\(23\!\cdots\!58\)\( T^{4} - \)\(16\!\cdots\!72\)\( T^{5} - \)\(14\!\cdots\!32\)\( T^{6} + \)\(34\!\cdots\!40\)\( T^{7} - \)\(62\!\cdots\!93\)\( T^{8} + \)\(34\!\cdots\!40\)\( p^{7} T^{9} - \)\(14\!\cdots\!32\)\( p^{14} T^{10} - \)\(16\!\cdots\!72\)\( p^{21} T^{11} + \)\(23\!\cdots\!58\)\( p^{28} T^{12} + 4801891406924128 p^{35} T^{13} - 21465016380 p^{42} T^{14} + 227504 p^{49} T^{15} + p^{56} T^{16} \)
37 \( ( 1 - 523580 T + 359904908818 T^{2} - 143318985284154800 T^{3} + \)\(13\!\cdots\!83\)\( p T^{4} - 143318985284154800 p^{7} T^{5} + 359904908818 p^{14} T^{6} - 523580 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
41 \( 1 - 67200 T - 515575757828 T^{2} + 39340714365093120 T^{3} + \)\(14\!\cdots\!54\)\( T^{4} - \)\(94\!\cdots\!40\)\( T^{5} - \)\(25\!\cdots\!24\)\( T^{6} + \)\(97\!\cdots\!60\)\( T^{7} + \)\(43\!\cdots\!55\)\( T^{8} + \)\(97\!\cdots\!60\)\( p^{7} T^{9} - \)\(25\!\cdots\!24\)\( p^{14} T^{10} - \)\(94\!\cdots\!40\)\( p^{21} T^{11} + \)\(14\!\cdots\!54\)\( p^{28} T^{12} + 39340714365093120 p^{35} T^{13} - 515575757828 p^{42} T^{14} - 67200 p^{49} T^{15} + p^{56} T^{16} \)
43 \( 1 + 562640 T - 420776333892 T^{2} - 138346669673028320 T^{3} + \)\(15\!\cdots\!78\)\( T^{4} + \)\(16\!\cdots\!20\)\( T^{5} - \)\(29\!\cdots\!96\)\( T^{6} - \)\(43\!\cdots\!60\)\( T^{7} + \)\(82\!\cdots\!59\)\( T^{8} - \)\(43\!\cdots\!60\)\( p^{7} T^{9} - \)\(29\!\cdots\!96\)\( p^{14} T^{10} + \)\(16\!\cdots\!20\)\( p^{21} T^{11} + \)\(15\!\cdots\!78\)\( p^{28} T^{12} - 138346669673028320 p^{35} T^{13} - 420776333892 p^{42} T^{14} + 562640 p^{49} T^{15} + p^{56} T^{16} \)
47 \( 1 - 515328 T - 1041981772508 T^{2} + 616082234310551040 T^{3} + \)\(45\!\cdots\!02\)\( T^{4} - \)\(21\!\cdots\!12\)\( T^{5} - \)\(29\!\cdots\!48\)\( T^{6} + \)\(17\!\cdots\!72\)\( T^{7} + \)\(21\!\cdots\!31\)\( T^{8} + \)\(17\!\cdots\!72\)\( p^{7} T^{9} - \)\(29\!\cdots\!48\)\( p^{14} T^{10} - \)\(21\!\cdots\!12\)\( p^{21} T^{11} + \)\(45\!\cdots\!02\)\( p^{28} T^{12} + 616082234310551040 p^{35} T^{13} - 1041981772508 p^{42} T^{14} - 515328 p^{49} T^{15} + p^{56} T^{16} \)
53 \( ( 1 + 2498016 T + 4667252414900 T^{2} + 6949193609921122080 T^{3} + \)\(86\!\cdots\!98\)\( T^{4} + 6949193609921122080 p^{7} T^{5} + 4667252414900 p^{14} T^{6} + 2498016 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
59 \( 1 - 4155840 T + 1386493873972 T^{2} + 3148704644236145280 T^{3} + \)\(41\!\cdots\!18\)\( T^{4} - \)\(74\!\cdots\!20\)\( T^{5} - \)\(36\!\cdots\!32\)\( T^{6} - \)\(71\!\cdots\!80\)\( T^{7} + \)\(50\!\cdots\!11\)\( T^{8} - \)\(71\!\cdots\!80\)\( p^{7} T^{9} - \)\(36\!\cdots\!32\)\( p^{14} T^{10} - \)\(74\!\cdots\!20\)\( p^{21} T^{11} + \)\(41\!\cdots\!18\)\( p^{28} T^{12} + 3148704644236145280 p^{35} T^{13} + 1386493873972 p^{42} T^{14} - 4155840 p^{49} T^{15} + p^{56} T^{16} \)
61 \( 1 + 2130764 T - 3014625418578 T^{2} + 2642708038716008248 T^{3} + \)\(20\!\cdots\!45\)\( T^{4} - \)\(21\!\cdots\!40\)\( T^{5} - \)\(17\!\cdots\!42\)\( T^{6} + \)\(53\!\cdots\!88\)\( T^{7} - \)\(35\!\cdots\!16\)\( T^{8} + \)\(53\!\cdots\!88\)\( p^{7} T^{9} - \)\(17\!\cdots\!42\)\( p^{14} T^{10} - \)\(21\!\cdots\!40\)\( p^{21} T^{11} + \)\(20\!\cdots\!45\)\( p^{28} T^{12} + 2642708038716008248 p^{35} T^{13} - 3014625418578 p^{42} T^{14} + 2130764 p^{49} T^{15} + p^{56} T^{16} \)
67 \( 1 - 1205440 T - 3106269411588 T^{2} + 35326918145455542400 T^{3} - \)\(33\!\cdots\!62\)\( T^{4} - \)\(14\!\cdots\!80\)\( T^{5} + \)\(46\!\cdots\!76\)\( T^{6} + \)\(50\!\cdots\!60\)\( T^{7} - \)\(31\!\cdots\!81\)\( T^{8} + \)\(50\!\cdots\!60\)\( p^{7} T^{9} + \)\(46\!\cdots\!76\)\( p^{14} T^{10} - \)\(14\!\cdots\!80\)\( p^{21} T^{11} - \)\(33\!\cdots\!62\)\( p^{28} T^{12} + 35326918145455542400 p^{35} T^{13} - 3106269411588 p^{42} T^{14} - 1205440 p^{49} T^{15} + p^{56} T^{16} \)
71 \( ( 1 + 12486480 T + 89708735416388 T^{2} + \)\(43\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!98\)\( T^{4} + \)\(43\!\cdots\!00\)\( p^{7} T^{5} + 89708735416388 p^{14} T^{6} + 12486480 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
73 \( ( 1 + 4820860 T + 38589071904874 T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + \)\(60\!\cdots\!87\)\( T^{4} + \)\(15\!\cdots\!40\)\( p^{7} T^{5} + 38589071904874 p^{14} T^{6} + 4820860 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
79 \( 1 - 11471680 T + 63121310835564 T^{2} - \)\(20\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!10\)\( T^{4} + \)\(41\!\cdots\!40\)\( p T^{5} - \)\(14\!\cdots\!64\)\( T^{6} + \)\(29\!\cdots\!20\)\( T^{7} - \)\(53\!\cdots\!41\)\( T^{8} + \)\(29\!\cdots\!20\)\( p^{7} T^{9} - \)\(14\!\cdots\!64\)\( p^{14} T^{10} + \)\(41\!\cdots\!40\)\( p^{22} T^{11} + \)\(11\!\cdots\!10\)\( p^{28} T^{12} - \)\(20\!\cdots\!60\)\( p^{35} T^{13} + 63121310835564 p^{42} T^{14} - 11471680 p^{49} T^{15} + p^{56} T^{16} \)
83 \( 1 - 16811232 T + 150404417986132 T^{2} - \)\(60\!\cdots\!64\)\( T^{3} - \)\(72\!\cdots\!22\)\( T^{4} + \)\(24\!\cdots\!24\)\( T^{5} - \)\(12\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!48\)\( T^{7} + \)\(88\!\cdots\!23\)\( T^{8} + \)\(14\!\cdots\!48\)\( p^{7} T^{9} - \)\(12\!\cdots\!76\)\( p^{14} T^{10} + \)\(24\!\cdots\!24\)\( p^{21} T^{11} - \)\(72\!\cdots\!22\)\( p^{28} T^{12} - \)\(60\!\cdots\!64\)\( p^{35} T^{13} + 150404417986132 p^{42} T^{14} - 16811232 p^{49} T^{15} + p^{56} T^{16} \)
89 \( ( 1 + 16857600 T + 212149846489550 T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!99\)\( T^{4} + \)\(19\!\cdots\!00\)\( p^{7} T^{5} + 212149846489550 p^{14} T^{6} + 16857600 p^{21} T^{7} + p^{28} T^{8} )^{2} \)
97 \( 1 - 27078040 T + 195301397964900 T^{2} - \)\(83\!\cdots\!20\)\( T^{3} + \)\(31\!\cdots\!74\)\( T^{4} - \)\(33\!\cdots\!40\)\( T^{5} + \)\(22\!\cdots\!00\)\( T^{6} - \)\(89\!\cdots\!20\)\( T^{7} + \)\(26\!\cdots\!15\)\( T^{8} - \)\(89\!\cdots\!20\)\( p^{7} T^{9} + \)\(22\!\cdots\!00\)\( p^{14} T^{10} - \)\(33\!\cdots\!40\)\( p^{21} T^{11} + \)\(31\!\cdots\!74\)\( p^{28} T^{12} - \)\(83\!\cdots\!20\)\( p^{35} T^{13} + 195301397964900 p^{42} T^{14} - 27078040 p^{49} T^{15} + p^{56} T^{16} \)
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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

−4.50504424329623490515450192546, −4.38926925138037015937677794882, −4.18668662303306678169518848810, −4.13961924688724109979990293844, −4.04308560338246013297939619833, −3.65247478255503966023656173234, −3.27604749545095963834656968884, −3.22574708571603623419571666206, −3.10435178624879395289770559386, −3.09106395947204579551871230648, −2.94831401367897386496884025960, −2.81141584589406002158750912230, −2.49428512813223368551869633917, −2.43849562915288251758386663072, −2.06601277547414477370042571938, −2.00234965120813209855034391298, −1.96569606221471804604086639996, −1.45259427237078618612190371517, −1.33189422800288319429391086216, −1.18521147330117219512521880638, −1.12164460266218258302419113557, −0.62051175249755717676729491892, −0.50555588315475018893684396031, −0.35631939352022218605034981868, −0.19180994383781480896047004425, 0.19180994383781480896047004425, 0.35631939352022218605034981868, 0.50555588315475018893684396031, 0.62051175249755717676729491892, 1.12164460266218258302419113557, 1.18521147330117219512521880638, 1.33189422800288319429391086216, 1.45259427237078618612190371517, 1.96569606221471804604086639996, 2.00234965120813209855034391298, 2.06601277547414477370042571938, 2.43849562915288251758386663072, 2.49428512813223368551869633917, 2.81141584589406002158750912230, 2.94831401367897386496884025960, 3.09106395947204579551871230648, 3.10435178624879395289770559386, 3.22574708571603623419571666206, 3.27604749545095963834656968884, 3.65247478255503966023656173234, 4.04308560338246013297939619833, 4.13961924688724109979990293844, 4.18668662303306678169518848810, 4.38926925138037015937677794882, 4.50504424329623490515450192546

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

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