Properties

Label 16-14e8-1.1-c13e8-0-0
Degree $16$
Conductor $1475789056$
Sign $1$
Analytic cond. $2.57979\times 10^{9}$
Root an. cond. $3.87457$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 256·2-s + 182·3-s + 2.45e4·4-s + 1.79e3·5-s + 4.65e4·6-s − 2.68e5·7-s + 3.50e6·9-s + 4.58e5·10-s − 8.72e6·11-s + 4.47e6·12-s + 1.43e6·13-s − 6.86e7·14-s + 3.26e5·15-s − 2.51e8·16-s − 7.94e6·17-s + 8.97e8·18-s − 2.15e8·19-s + 4.40e7·20-s − 4.88e7·21-s − 2.23e9·22-s − 6.19e7·23-s + 1.77e9·25-s + 3.68e8·26-s − 2.60e8·27-s − 6.59e9·28-s − 6.32e9·29-s + 8.34e7·30-s + ⋯
L(s)  = 1  + 2.82·2-s + 0.144·3-s + 3·4-s + 0.0512·5-s + 0.407·6-s − 0.862·7-s + 2.19·9-s + 0.145·10-s − 1.48·11-s + 0.432·12-s + 0.0826·13-s − 2.43·14-s + 0.00739·15-s − 3.75·16-s − 0.0798·17-s + 6.21·18-s − 1.05·19-s + 0.153·20-s − 0.124·21-s − 4.20·22-s − 0.0872·23-s + 1.45·25-s + 0.233·26-s − 0.129·27-s − 2.58·28-s − 1.97·29-s + 0.0209·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.57979\times 10^{9}\)
Root analytic conductor: \(3.87457\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{14} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 7^{8} ,\ ( \ : [13/2]^{8} ),\ 1 )\)

Particular Values

\(L(7)\) \(\approx\) \(0.2344700904\)
\(L(\frac12)\) \(\approx\) \(0.2344700904\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p^{6} T + p^{12} T^{2} )^{4} \)
7 \( 1 + 38336 p T - 115258988 p^{3} T^{2} - 1173216704 p^{8} T^{3} + 5981537894 p^{14} T^{4} - 1173216704 p^{21} T^{5} - 115258988 p^{29} T^{6} + 38336 p^{40} T^{7} + p^{52} T^{8} \)
good3 \( 1 - 182 T - 3472004 T^{2} + 510005888 p T^{3} + 610111576351 p^{2} T^{4} - 38594331621020 p^{4} T^{5} - 6098702794604744 p^{6} T^{6} + 132316912065600514 p^{9} T^{7} + 7966403831993958904 p^{12} T^{8} + 132316912065600514 p^{22} T^{9} - 6098702794604744 p^{32} T^{10} - 38594331621020 p^{43} T^{11} + 610111576351 p^{54} T^{12} + 510005888 p^{66} T^{13} - 3472004 p^{78} T^{14} - 182 p^{91} T^{15} + p^{104} T^{16} \)
5 \( 1 - 1792 T - 1775878222 T^{2} + 184747900060528 T^{3} + 2296434795996497581 T^{4} - \)\(55\!\cdots\!84\)\( p T^{5} + \)\(50\!\cdots\!26\)\( p^{2} T^{6} + \)\(53\!\cdots\!12\)\( p^{4} T^{7} - \)\(11\!\cdots\!96\)\( p^{6} T^{8} + \)\(53\!\cdots\!12\)\( p^{17} T^{9} + \)\(50\!\cdots\!26\)\( p^{28} T^{10} - \)\(55\!\cdots\!84\)\( p^{40} T^{11} + 2296434795996497581 p^{52} T^{12} + 184747900060528 p^{65} T^{13} - 1775878222 p^{78} T^{14} - 1792 p^{91} T^{15} + p^{104} T^{16} \)
11 \( 1 + 8726914 T - 25911547427740 T^{2} - 67039793147424510224 p T^{3} - \)\(11\!\cdots\!25\)\( p^{2} T^{4} + \)\(22\!\cdots\!28\)\( p^{3} T^{5} + \)\(12\!\cdots\!92\)\( p^{4} T^{6} - \)\(31\!\cdots\!90\)\( p^{5} T^{7} - \)\(47\!\cdots\!56\)\( p^{6} T^{8} - \)\(31\!\cdots\!90\)\( p^{18} T^{9} + \)\(12\!\cdots\!92\)\( p^{30} T^{10} + \)\(22\!\cdots\!28\)\( p^{42} T^{11} - \)\(11\!\cdots\!25\)\( p^{54} T^{12} - 67039793147424510224 p^{66} T^{13} - 25911547427740 p^{78} T^{14} + 8726914 p^{91} T^{15} + p^{104} T^{16} \)
13 \( ( 1 - 719208 T + 661703400953692 T^{2} - \)\(30\!\cdots\!92\)\( T^{3} + \)\(24\!\cdots\!34\)\( T^{4} - \)\(30\!\cdots\!92\)\( p^{13} T^{5} + 661703400953692 p^{26} T^{6} - 719208 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
17 \( 1 + 7943068 T - 12420153001480906 T^{2} + \)\(15\!\cdots\!36\)\( T^{3} + \)\(32\!\cdots\!53\)\( T^{4} - \)\(18\!\cdots\!12\)\( T^{5} + \)\(14\!\cdots\!82\)\( T^{6} + \)\(10\!\cdots\!96\)\( T^{7} - \)\(13\!\cdots\!44\)\( T^{8} + \)\(10\!\cdots\!96\)\( p^{13} T^{9} + \)\(14\!\cdots\!82\)\( p^{26} T^{10} - \)\(18\!\cdots\!12\)\( p^{39} T^{11} + \)\(32\!\cdots\!53\)\( p^{52} T^{12} + \)\(15\!\cdots\!36\)\( p^{65} T^{13} - 12420153001480906 p^{78} T^{14} + 7943068 p^{91} T^{15} + p^{104} T^{16} \)
19 \( 1 + 215706806 T - 83649252740873580 T^{2} - \)\(64\!\cdots\!84\)\( T^{3} + \)\(61\!\cdots\!95\)\( T^{4} - \)\(73\!\cdots\!52\)\( T^{5} - \)\(31\!\cdots\!04\)\( T^{6} - \)\(14\!\cdots\!30\)\( T^{7} + \)\(11\!\cdots\!32\)\( T^{8} - \)\(14\!\cdots\!30\)\( p^{13} T^{9} - \)\(31\!\cdots\!04\)\( p^{26} T^{10} - \)\(73\!\cdots\!52\)\( p^{39} T^{11} + \)\(61\!\cdots\!95\)\( p^{52} T^{12} - \)\(64\!\cdots\!84\)\( p^{65} T^{13} - 83649252740873580 p^{78} T^{14} + 215706806 p^{91} T^{15} + p^{104} T^{16} \)
23 \( 1 + 61927978 T - 1190855939592873856 T^{2} + \)\(30\!\cdots\!76\)\( T^{3} + \)\(72\!\cdots\!19\)\( T^{4} - \)\(28\!\cdots\!84\)\( T^{5} - \)\(22\!\cdots\!24\)\( T^{6} + \)\(84\!\cdots\!66\)\( T^{7} + \)\(67\!\cdots\!48\)\( T^{8} + \)\(84\!\cdots\!66\)\( p^{13} T^{9} - \)\(22\!\cdots\!24\)\( p^{26} T^{10} - \)\(28\!\cdots\!84\)\( p^{39} T^{11} + \)\(72\!\cdots\!19\)\( p^{52} T^{12} + \)\(30\!\cdots\!76\)\( p^{65} T^{13} - 1190855939592873856 p^{78} T^{14} + 61927978 p^{91} T^{15} + p^{104} T^{16} \)
29 \( ( 1 + 3162923032 T + 22916063388720489116 T^{2} + \)\(10\!\cdots\!52\)\( T^{3} + \)\(16\!\cdots\!10\)\( T^{4} + \)\(10\!\cdots\!52\)\( p^{13} T^{5} + 22916063388720489116 p^{26} T^{6} + 3162923032 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
31 \( 1 - 6113775570 T - 34184846727810893416 T^{2} - \)\(15\!\cdots\!56\)\( T^{3} + \)\(19\!\cdots\!11\)\( T^{4} + \)\(31\!\cdots\!76\)\( T^{5} - \)\(38\!\cdots\!04\)\( T^{6} - \)\(15\!\cdots\!66\)\( T^{7} + \)\(24\!\cdots\!44\)\( T^{8} - \)\(15\!\cdots\!66\)\( p^{13} T^{9} - \)\(38\!\cdots\!04\)\( p^{26} T^{10} + \)\(31\!\cdots\!76\)\( p^{39} T^{11} + \)\(19\!\cdots\!11\)\( p^{52} T^{12} - \)\(15\!\cdots\!56\)\( p^{65} T^{13} - 34184846727810893416 p^{78} T^{14} - 6113775570 p^{91} T^{15} + p^{104} T^{16} \)
37 \( 1 + 3945652880 T - \)\(35\!\cdots\!22\)\( T^{2} - \)\(75\!\cdots\!84\)\( T^{3} + \)\(29\!\cdots\!37\)\( T^{4} + \)\(20\!\cdots\!48\)\( T^{5} + \)\(26\!\cdots\!46\)\( T^{6} - \)\(26\!\cdots\!52\)\( T^{7} - \)\(78\!\cdots\!16\)\( T^{8} - \)\(26\!\cdots\!52\)\( p^{13} T^{9} + \)\(26\!\cdots\!46\)\( p^{26} T^{10} + \)\(20\!\cdots\!48\)\( p^{39} T^{11} + \)\(29\!\cdots\!37\)\( p^{52} T^{12} - \)\(75\!\cdots\!84\)\( p^{65} T^{13} - \)\(35\!\cdots\!22\)\( p^{78} T^{14} + 3945652880 p^{91} T^{15} + p^{104} T^{16} \)
41 \( ( 1 - 43189289976 T + \)\(25\!\cdots\!84\)\( T^{2} - \)\(40\!\cdots\!84\)\( T^{3} + \)\(19\!\cdots\!30\)\( T^{4} - \)\(40\!\cdots\!84\)\( p^{13} T^{5} + \)\(25\!\cdots\!84\)\( p^{26} T^{6} - 43189289976 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
43 \( ( 1 + 54537062128 T + \)\(53\!\cdots\!52\)\( T^{2} + \)\(27\!\cdots\!32\)\( T^{3} + \)\(12\!\cdots\!74\)\( T^{4} + \)\(27\!\cdots\!32\)\( p^{13} T^{5} + \)\(53\!\cdots\!52\)\( p^{26} T^{6} + 54537062128 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
47 \( 1 + 3141202722 T - \)\(99\!\cdots\!08\)\( T^{2} + \)\(96\!\cdots\!24\)\( T^{3} + \)\(56\!\cdots\!43\)\( T^{4} - \)\(74\!\cdots\!24\)\( T^{5} + \)\(37\!\cdots\!04\)\( T^{6} + \)\(31\!\cdots\!22\)\( T^{7} - \)\(30\!\cdots\!12\)\( T^{8} + \)\(31\!\cdots\!22\)\( p^{13} T^{9} + \)\(37\!\cdots\!04\)\( p^{26} T^{10} - \)\(74\!\cdots\!24\)\( p^{39} T^{11} + \)\(56\!\cdots\!43\)\( p^{52} T^{12} + \)\(96\!\cdots\!24\)\( p^{65} T^{13} - \)\(99\!\cdots\!08\)\( p^{78} T^{14} + 3141202722 p^{91} T^{15} + p^{104} T^{16} \)
53 \( 1 - 149625680376 T - \)\(32\!\cdots\!14\)\( T^{2} + \)\(30\!\cdots\!72\)\( T^{3} + \)\(35\!\cdots\!73\)\( T^{4} + \)\(83\!\cdots\!64\)\( T^{5} - \)\(14\!\cdots\!22\)\( T^{6} - \)\(22\!\cdots\!88\)\( T^{7} + \)\(70\!\cdots\!56\)\( T^{8} - \)\(22\!\cdots\!88\)\( p^{13} T^{9} - \)\(14\!\cdots\!22\)\( p^{26} T^{10} + \)\(83\!\cdots\!64\)\( p^{39} T^{11} + \)\(35\!\cdots\!73\)\( p^{52} T^{12} + \)\(30\!\cdots\!72\)\( p^{65} T^{13} - \)\(32\!\cdots\!14\)\( p^{78} T^{14} - 149625680376 p^{91} T^{15} + p^{104} T^{16} \)
59 \( 1 + 866297313938 T + \)\(26\!\cdots\!16\)\( T^{2} + \)\(32\!\cdots\!72\)\( T^{3} - \)\(16\!\cdots\!01\)\( T^{4} - \)\(55\!\cdots\!40\)\( T^{5} - \)\(29\!\cdots\!16\)\( T^{6} - \)\(69\!\cdots\!46\)\( T^{7} - \)\(13\!\cdots\!88\)\( T^{8} - \)\(69\!\cdots\!46\)\( p^{13} T^{9} - \)\(29\!\cdots\!16\)\( p^{26} T^{10} - \)\(55\!\cdots\!40\)\( p^{39} T^{11} - \)\(16\!\cdots\!01\)\( p^{52} T^{12} + \)\(32\!\cdots\!72\)\( p^{65} T^{13} + \)\(26\!\cdots\!16\)\( p^{78} T^{14} + 866297313938 p^{91} T^{15} + p^{104} T^{16} \)
61 \( 1 + 477908594184 T - \)\(49\!\cdots\!54\)\( T^{2} - \)\(12\!\cdots\!32\)\( T^{3} + \)\(22\!\cdots\!57\)\( T^{4} + \)\(35\!\cdots\!84\)\( T^{5} - \)\(56\!\cdots\!18\)\( T^{6} - \)\(12\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!56\)\( T^{8} - \)\(12\!\cdots\!52\)\( p^{13} T^{9} - \)\(56\!\cdots\!18\)\( p^{26} T^{10} + \)\(35\!\cdots\!84\)\( p^{39} T^{11} + \)\(22\!\cdots\!57\)\( p^{52} T^{12} - \)\(12\!\cdots\!32\)\( p^{65} T^{13} - \)\(49\!\cdots\!54\)\( p^{78} T^{14} + 477908594184 p^{91} T^{15} + p^{104} T^{16} \)
67 \( 1 - 1895501016278 T + \)\(16\!\cdots\!44\)\( T^{2} + \)\(36\!\cdots\!92\)\( T^{3} + \)\(20\!\cdots\!99\)\( T^{4} - \)\(15\!\cdots\!64\)\( T^{5} - \)\(63\!\cdots\!80\)\( T^{6} - \)\(28\!\cdots\!02\)\( T^{7} + \)\(12\!\cdots\!00\)\( T^{8} - \)\(28\!\cdots\!02\)\( p^{13} T^{9} - \)\(63\!\cdots\!80\)\( p^{26} T^{10} - \)\(15\!\cdots\!64\)\( p^{39} T^{11} + \)\(20\!\cdots\!99\)\( p^{52} T^{12} + \)\(36\!\cdots\!92\)\( p^{65} T^{13} + \)\(16\!\cdots\!44\)\( p^{78} T^{14} - 1895501016278 p^{91} T^{15} + p^{104} T^{16} \)
71 \( ( 1 - 319416336064 T + \)\(11\!\cdots\!92\)\( T^{2} + \)\(18\!\cdots\!52\)\( T^{3} + \)\(28\!\cdots\!78\)\( p T^{4} + \)\(18\!\cdots\!52\)\( p^{13} T^{5} + \)\(11\!\cdots\!92\)\( p^{26} T^{6} - 319416336064 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
73 \( 1 + 2966596192756 T - \)\(63\!\cdots\!50\)\( T^{2} - \)\(36\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!25\)\( T^{4} + \)\(21\!\cdots\!56\)\( T^{5} - \)\(21\!\cdots\!70\)\( T^{6} + \)\(68\!\cdots\!04\)\( p T^{7} + \)\(16\!\cdots\!24\)\( p^{2} T^{8} + \)\(68\!\cdots\!04\)\( p^{14} T^{9} - \)\(21\!\cdots\!70\)\( p^{26} T^{10} + \)\(21\!\cdots\!56\)\( p^{39} T^{11} + \)\(16\!\cdots\!25\)\( p^{52} T^{12} - \)\(36\!\cdots\!60\)\( p^{65} T^{13} - \)\(63\!\cdots\!50\)\( p^{78} T^{14} + 2966596192756 p^{91} T^{15} + p^{104} T^{16} \)
79 \( 1 + 6505959677634 T + \)\(10\!\cdots\!40\)\( T^{2} + \)\(11\!\cdots\!68\)\( T^{3} + \)\(13\!\cdots\!83\)\( T^{4} + \)\(31\!\cdots\!32\)\( T^{5} - \)\(17\!\cdots\!68\)\( T^{6} + \)\(60\!\cdots\!74\)\( T^{7} + \)\(43\!\cdots\!56\)\( T^{8} + \)\(60\!\cdots\!74\)\( p^{13} T^{9} - \)\(17\!\cdots\!68\)\( p^{26} T^{10} + \)\(31\!\cdots\!32\)\( p^{39} T^{11} + \)\(13\!\cdots\!83\)\( p^{52} T^{12} + \)\(11\!\cdots\!68\)\( p^{65} T^{13} + \)\(10\!\cdots\!40\)\( p^{78} T^{14} + 6505959677634 p^{91} T^{15} + p^{104} T^{16} \)
83 \( ( 1 + 1689908567984 T + \)\(20\!\cdots\!40\)\( T^{2} + \)\(40\!\cdots\!32\)\( T^{3} + \)\(20\!\cdots\!38\)\( T^{4} + \)\(40\!\cdots\!32\)\( p^{13} T^{5} + \)\(20\!\cdots\!40\)\( p^{26} T^{6} + 1689908567984 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
89 \( 1 - 9586601667468 T - \)\(11\!\cdots\!14\)\( T^{2} + \)\(13\!\cdots\!04\)\( T^{3} + \)\(18\!\cdots\!49\)\( T^{4} - \)\(53\!\cdots\!00\)\( T^{5} - \)\(54\!\cdots\!90\)\( T^{6} + \)\(55\!\cdots\!68\)\( T^{7} + \)\(12\!\cdots\!52\)\( T^{8} + \)\(55\!\cdots\!68\)\( p^{13} T^{9} - \)\(54\!\cdots\!90\)\( p^{26} T^{10} - \)\(53\!\cdots\!00\)\( p^{39} T^{11} + \)\(18\!\cdots\!49\)\( p^{52} T^{12} + \)\(13\!\cdots\!04\)\( p^{65} T^{13} - \)\(11\!\cdots\!14\)\( p^{78} T^{14} - 9586601667468 p^{91} T^{15} + p^{104} T^{16} \)
97 \( ( 1 + 22280367655784 T + \)\(39\!\cdots\!28\)\( T^{2} + \)\(43\!\cdots\!36\)\( T^{3} + \)\(42\!\cdots\!74\)\( T^{4} + \)\(43\!\cdots\!36\)\( p^{13} T^{5} + \)\(39\!\cdots\!28\)\( p^{26} T^{6} + 22280367655784 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
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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

−6.36765821679055246556175587690, −5.98511076326541554076456885122, −5.91651713318611611728478057526, −5.66307475700102631451840109785, −5.65867954484030400555419836781, −5.03146044966849350581870089078, −4.73934187318295763589757543889, −4.73571821817269864954902540026, −4.60824578979732840832767291432, −4.49316695987840878654794507909, −4.12014752295418179011544794237, −3.65368761205294173122189503424, −3.63402336398901053449189313561, −3.55525541228603105073194405151, −3.16670027599768625710971812208, −2.82379028073472069892657516563, −2.54738380356064688896284778271, −2.43208766918718112815508495584, −2.10765151307443108673067106960, −1.61242001388358858830932104211, −1.54506551789164061948829893906, −1.03825390696610814602550554896, −0.792605522114908835596061840747, −0.38537465632742703949773636713, −0.02927334630808015149752848512, 0.02927334630808015149752848512, 0.38537465632742703949773636713, 0.792605522114908835596061840747, 1.03825390696610814602550554896, 1.54506551789164061948829893906, 1.61242001388358858830932104211, 2.10765151307443108673067106960, 2.43208766918718112815508495584, 2.54738380356064688896284778271, 2.82379028073472069892657516563, 3.16670027599768625710971812208, 3.55525541228603105073194405151, 3.63402336398901053449189313561, 3.65368761205294173122189503424, 4.12014752295418179011544794237, 4.49316695987840878654794507909, 4.60824578979732840832767291432, 4.73571821817269864954902540026, 4.73934187318295763589757543889, 5.03146044966849350581870089078, 5.65867954484030400555419836781, 5.66307475700102631451840109785, 5.91651713318611611728478057526, 5.98511076326541554076456885122, 6.36765821679055246556175587690

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

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