Properties

Label 16-688e8-1.1-c5e8-0-1
Degree $16$
Conductor $5.020\times 10^{22}$
Sign $1$
Analytic cond. $2.19781\times 10^{16}$
Root an. cond. $10.5044$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $8$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 26·3-s − 212·5-s + 136·7-s − 361·9-s + 532·11-s − 2.49e3·13-s − 5.51e3·15-s − 2.53e3·17-s + 1.67e3·19-s + 3.53e3·21-s + 2.48e3·23-s + 1.21e4·25-s − 1.43e4·27-s − 4.36e3·29-s − 5.70e3·31-s + 1.38e4·33-s − 2.88e4·35-s − 3.77e3·37-s − 6.47e4·39-s − 1.06e4·41-s + 1.47e4·43-s + 7.65e4·45-s + 7.78e4·47-s − 5.43e4·49-s − 6.58e4·51-s − 6.23e4·53-s − 1.12e5·55-s + ⋯
L(s)  = 1  + 1.66·3-s − 3.79·5-s + 1.04·7-s − 1.48·9-s + 1.32·11-s − 4.08·13-s − 6.32·15-s − 2.12·17-s + 1.06·19-s + 1.74·21-s + 0.980·23-s + 3.89·25-s − 3.79·27-s − 0.962·29-s − 1.06·31-s + 2.21·33-s − 3.97·35-s − 0.452·37-s − 6.82·39-s − 0.993·41-s + 1.21·43-s + 5.63·45-s + 5.14·47-s − 3.23·49-s − 3.54·51-s − 3.04·53-s − 5.02·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 43^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 43^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 43^{8}\)
Sign: $1$
Analytic conductor: \(2.19781\times 10^{16}\)
Root analytic conductor: \(10.5044\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 2^{32} \cdot 43^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( ( 1 - p^{2} T )^{8} \)
good3 \( 1 - 26 T + 1037 T^{2} - 7328 p T^{3} + 515401 T^{4} - 8844604 T^{5} + 164541593 T^{6} - 804076450 p T^{7} + 4709963816 p^{2} T^{8} - 804076450 p^{6} T^{9} + 164541593 p^{10} T^{10} - 8844604 p^{15} T^{11} + 515401 p^{20} T^{12} - 7328 p^{26} T^{13} + 1037 p^{30} T^{14} - 26 p^{35} T^{15} + p^{40} T^{16} \)
5 \( 1 + 212 T + 32783 T^{2} + 3621372 T^{3} + 67597473 p T^{4} + 26716499536 T^{5} + 1886184834567 T^{6} + 120009344570848 T^{7} + 7002693082424752 T^{8} + 120009344570848 p^{5} T^{9} + 1886184834567 p^{10} T^{10} + 26716499536 p^{15} T^{11} + 67597473 p^{21} T^{12} + 3621372 p^{25} T^{13} + 32783 p^{30} T^{14} + 212 p^{35} T^{15} + p^{40} T^{16} \)
7 \( 1 - 136 T + 72882 T^{2} - 890912 p T^{3} + 2273171472 T^{4} - 116085882896 T^{5} + 47825461769326 T^{6} - 1608183672345720 T^{7} + 855930962462754846 T^{8} - 1608183672345720 p^{5} T^{9} + 47825461769326 p^{10} T^{10} - 116085882896 p^{15} T^{11} + 2273171472 p^{20} T^{12} - 890912 p^{26} T^{13} + 72882 p^{30} T^{14} - 136 p^{35} T^{15} + p^{40} T^{16} \)
11 \( 1 - 532 T + 730390 T^{2} - 277407524 T^{3} + 233648426873 T^{4} - 75202699287360 T^{5} + 55193097116019702 T^{6} - 16538641578891295312 T^{7} + \)\(10\!\cdots\!12\)\( T^{8} - 16538641578891295312 p^{5} T^{9} + 55193097116019702 p^{10} T^{10} - 75202699287360 p^{15} T^{11} + 233648426873 p^{20} T^{12} - 277407524 p^{25} T^{13} + 730390 p^{30} T^{14} - 532 p^{35} T^{15} + p^{40} T^{16} \)
13 \( 1 + 2492 T + 4165454 T^{2} + 5101254988 T^{3} + 390512688221 p T^{4} + 4290072329984560 T^{5} + 3197308617811643622 T^{6} + \)\(21\!\cdots\!28\)\( T^{7} + \)\(13\!\cdots\!48\)\( T^{8} + \)\(21\!\cdots\!28\)\( p^{5} T^{9} + 3197308617811643622 p^{10} T^{10} + 4290072329984560 p^{15} T^{11} + 390512688221 p^{21} T^{12} + 5101254988 p^{25} T^{13} + 4165454 p^{30} T^{14} + 2492 p^{35} T^{15} + p^{40} T^{16} \)
17 \( 1 + 2534 T + 10088213 T^{2} + 18871627772 T^{3} + 43872695037958 T^{4} + 64773449524240486 T^{5} + \)\(11\!\cdots\!72\)\( T^{6} + \)\(13\!\cdots\!98\)\( T^{7} + \)\(19\!\cdots\!73\)\( T^{8} + \)\(13\!\cdots\!98\)\( p^{5} T^{9} + \)\(11\!\cdots\!72\)\( p^{10} T^{10} + 64773449524240486 p^{15} T^{11} + 43872695037958 p^{20} T^{12} + 18871627772 p^{25} T^{13} + 10088213 p^{30} T^{14} + 2534 p^{35} T^{15} + p^{40} T^{16} \)
19 \( 1 - 1678 T + 4805907 T^{2} - 3688923072 T^{3} + 6298623964783 T^{4} - 4799890940703308 T^{5} + 9212340221857328609 T^{6} - \)\(43\!\cdots\!14\)\( T^{7} + \)\(59\!\cdots\!68\)\( T^{8} - \)\(43\!\cdots\!14\)\( p^{5} T^{9} + 9212340221857328609 p^{10} T^{10} - 4799890940703308 p^{15} T^{11} + 6298623964783 p^{20} T^{12} - 3688923072 p^{25} T^{13} + 4805907 p^{30} T^{14} - 1678 p^{35} T^{15} + p^{40} T^{16} \)
23 \( 1 - 2488 T + 19807447 T^{2} - 41115937228 T^{3} + 232966325434474 T^{4} - 412893749849663568 T^{5} + \)\(20\!\cdots\!08\)\( T^{6} - \)\(32\!\cdots\!88\)\( T^{7} + \)\(14\!\cdots\!09\)\( T^{8} - \)\(32\!\cdots\!88\)\( p^{5} T^{9} + \)\(20\!\cdots\!08\)\( p^{10} T^{10} - 412893749849663568 p^{15} T^{11} + 232966325434474 p^{20} T^{12} - 41115937228 p^{25} T^{13} + 19807447 p^{30} T^{14} - 2488 p^{35} T^{15} + p^{40} T^{16} \)
29 \( 1 + 4360 T + 123420923 T^{2} + 388928438620 T^{3} + 6671975392676125 T^{4} + 14971808484524322944 T^{5} + \)\(21\!\cdots\!99\)\( T^{6} + \)\(36\!\cdots\!76\)\( T^{7} + \)\(51\!\cdots\!68\)\( T^{8} + \)\(36\!\cdots\!76\)\( p^{5} T^{9} + \)\(21\!\cdots\!99\)\( p^{10} T^{10} + 14971808484524322944 p^{15} T^{11} + 6671975392676125 p^{20} T^{12} + 388928438620 p^{25} T^{13} + 123420923 p^{30} T^{14} + 4360 p^{35} T^{15} + p^{40} T^{16} \)
31 \( 1 + 184 p T + 4985085 p T^{2} + 536298509268 T^{3} + 10208569817532730 T^{4} + 23021651844205871336 T^{5} + \)\(44\!\cdots\!56\)\( T^{6} + \)\(75\!\cdots\!68\)\( T^{7} + \)\(14\!\cdots\!33\)\( T^{8} + \)\(75\!\cdots\!68\)\( p^{5} T^{9} + \)\(44\!\cdots\!56\)\( p^{10} T^{10} + 23021651844205871336 p^{15} T^{11} + 10208569817532730 p^{20} T^{12} + 536298509268 p^{25} T^{13} + 4985085 p^{31} T^{14} + 184 p^{36} T^{15} + p^{40} T^{16} \)
37 \( 1 + 3772 T + 191073025 T^{2} - 279875011868 T^{3} + 17131267855690943 T^{4} - 65662976081645242692 T^{5} + \)\(16\!\cdots\!59\)\( T^{6} - \)\(36\!\cdots\!12\)\( T^{7} + \)\(13\!\cdots\!56\)\( T^{8} - \)\(36\!\cdots\!12\)\( p^{5} T^{9} + \)\(16\!\cdots\!59\)\( p^{10} T^{10} - 65662976081645242692 p^{15} T^{11} + 17131267855690943 p^{20} T^{12} - 279875011868 p^{25} T^{13} + 191073025 p^{30} T^{14} + 3772 p^{35} T^{15} + p^{40} T^{16} \)
41 \( 1 + 10698 T + 718082789 T^{2} + 6825101409152 T^{3} + 244397855377030766 T^{4} + \)\(20\!\cdots\!98\)\( T^{5} + \)\(51\!\cdots\!60\)\( T^{6} + \)\(36\!\cdots\!02\)\( T^{7} + \)\(71\!\cdots\!21\)\( T^{8} + \)\(36\!\cdots\!02\)\( p^{5} T^{9} + \)\(51\!\cdots\!60\)\( p^{10} T^{10} + \)\(20\!\cdots\!98\)\( p^{15} T^{11} + 244397855377030766 p^{20} T^{12} + 6825101409152 p^{25} T^{13} + 718082789 p^{30} T^{14} + 10698 p^{35} T^{15} + p^{40} T^{16} \)
47 \( 1 - 77864 T + 4209868025 T^{2} - 159292711218168 T^{3} + 4925605431015049275 T^{4} - \)\(12\!\cdots\!88\)\( T^{5} + \)\(26\!\cdots\!31\)\( T^{6} - \)\(50\!\cdots\!80\)\( T^{7} + \)\(81\!\cdots\!28\)\( T^{8} - \)\(50\!\cdots\!80\)\( p^{5} T^{9} + \)\(26\!\cdots\!31\)\( p^{10} T^{10} - \)\(12\!\cdots\!88\)\( p^{15} T^{11} + 4925605431015049275 p^{20} T^{12} - 159292711218168 p^{25} T^{13} + 4209868025 p^{30} T^{14} - 77864 p^{35} T^{15} + p^{40} T^{16} \)
53 \( 1 + 62352 T + 4104997514 T^{2} + 164980080105492 T^{3} + 6411159574721863281 T^{4} + \)\(19\!\cdots\!48\)\( T^{5} + \)\(54\!\cdots\!26\)\( T^{6} + \)\(12\!\cdots\!52\)\( T^{7} + \)\(28\!\cdots\!76\)\( T^{8} + \)\(12\!\cdots\!52\)\( p^{5} T^{9} + \)\(54\!\cdots\!26\)\( p^{10} T^{10} + \)\(19\!\cdots\!48\)\( p^{15} T^{11} + 6411159574721863281 p^{20} T^{12} + 164980080105492 p^{25} T^{13} + 4104997514 p^{30} T^{14} + 62352 p^{35} T^{15} + p^{40} T^{16} \)
59 \( 1 - 26224 T + 2774515012 T^{2} - 45419810356528 T^{3} + 3251157373342627972 T^{4} - \)\(32\!\cdots\!68\)\( T^{5} + \)\(23\!\cdots\!96\)\( T^{6} - \)\(11\!\cdots\!40\)\( T^{7} + \)\(15\!\cdots\!70\)\( T^{8} - \)\(11\!\cdots\!40\)\( p^{5} T^{9} + \)\(23\!\cdots\!96\)\( p^{10} T^{10} - \)\(32\!\cdots\!68\)\( p^{15} T^{11} + 3251157373342627972 p^{20} T^{12} - 45419810356528 p^{25} T^{13} + 2774515012 p^{30} T^{14} - 26224 p^{35} T^{15} + p^{40} T^{16} \)
61 \( 1 + 82540 T + 7066815930 T^{2} + 341557091503436 T^{3} + 16786861393791028176 T^{4} + \)\(57\!\cdots\!44\)\( T^{5} + \)\(21\!\cdots\!70\)\( T^{6} + \)\(60\!\cdots\!16\)\( T^{7} + \)\(19\!\cdots\!30\)\( T^{8} + \)\(60\!\cdots\!16\)\( p^{5} T^{9} + \)\(21\!\cdots\!70\)\( p^{10} T^{10} + \)\(57\!\cdots\!44\)\( p^{15} T^{11} + 16786861393791028176 p^{20} T^{12} + 341557091503436 p^{25} T^{13} + 7066815930 p^{30} T^{14} + 82540 p^{35} T^{15} + p^{40} T^{16} \)
67 \( 1 + 27784 T + 7986133086 T^{2} + 191439161437872 T^{3} + 30490262918267386393 T^{4} + \)\(62\!\cdots\!04\)\( T^{5} + \)\(72\!\cdots\!82\)\( T^{6} + \)\(12\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!76\)\( T^{8} + \)\(12\!\cdots\!60\)\( p^{5} T^{9} + \)\(72\!\cdots\!82\)\( p^{10} T^{10} + \)\(62\!\cdots\!04\)\( p^{15} T^{11} + 30490262918267386393 p^{20} T^{12} + 191439161437872 p^{25} T^{13} + 7986133086 p^{30} T^{14} + 27784 p^{35} T^{15} + p^{40} T^{16} \)
71 \( 1 - 9504 T + 6761837728 T^{2} - 123767778948544 T^{3} + 23139305175095415788 T^{4} - \)\(61\!\cdots\!44\)\( T^{5} + \)\(57\!\cdots\!36\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!86\)\( T^{8} - \)\(17\!\cdots\!00\)\( p^{5} T^{9} + \)\(57\!\cdots\!36\)\( p^{10} T^{10} - \)\(61\!\cdots\!44\)\( p^{15} T^{11} + 23139305175095415788 p^{20} T^{12} - 123767778948544 p^{25} T^{13} + 6761837728 p^{30} T^{14} - 9504 p^{35} T^{15} + p^{40} T^{16} \)
73 \( 1 - 14260 T + 14183423290 T^{2} - 183318998319396 T^{3} + 92012288750177195552 T^{4} - \)\(10\!\cdots\!48\)\( T^{5} + \)\(35\!\cdots\!10\)\( T^{6} - \)\(34\!\cdots\!56\)\( T^{7} + \)\(90\!\cdots\!14\)\( T^{8} - \)\(34\!\cdots\!56\)\( p^{5} T^{9} + \)\(35\!\cdots\!10\)\( p^{10} T^{10} - \)\(10\!\cdots\!48\)\( p^{15} T^{11} + 92012288750177195552 p^{20} T^{12} - 183318998319396 p^{25} T^{13} + 14183423290 p^{30} T^{14} - 14260 p^{35} T^{15} + p^{40} T^{16} \)
79 \( 1 + 160248 T + 28127571285 T^{2} + 2813803255473368 T^{3} + \)\(29\!\cdots\!07\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!27\)\( T^{6} + \)\(10\!\cdots\!60\)\( T^{7} + \)\(62\!\cdots\!72\)\( T^{8} + \)\(10\!\cdots\!60\)\( p^{5} T^{9} + \)\(16\!\cdots\!27\)\( p^{10} T^{10} + \)\(21\!\cdots\!00\)\( p^{15} T^{11} + \)\(29\!\cdots\!07\)\( p^{20} T^{12} + 2813803255473368 p^{25} T^{13} + 28127571285 p^{30} T^{14} + 160248 p^{35} T^{15} + p^{40} T^{16} \)
83 \( 1 - 77176 T + 20087963970 T^{2} - 1576267525049076 T^{3} + \)\(20\!\cdots\!93\)\( T^{4} - \)\(15\!\cdots\!84\)\( T^{5} + \)\(13\!\cdots\!78\)\( T^{6} - \)\(91\!\cdots\!20\)\( T^{7} + \)\(61\!\cdots\!64\)\( T^{8} - \)\(91\!\cdots\!20\)\( p^{5} T^{9} + \)\(13\!\cdots\!78\)\( p^{10} T^{10} - \)\(15\!\cdots\!84\)\( p^{15} T^{11} + \)\(20\!\cdots\!93\)\( p^{20} T^{12} - 1576267525049076 p^{25} T^{13} + 20087963970 p^{30} T^{14} - 77176 p^{35} T^{15} + p^{40} T^{16} \)
89 \( 1 + 265692 T + 60287015686 T^{2} + 9508315264777636 T^{3} + \)\(13\!\cdots\!36\)\( T^{4} + \)\(15\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!78\)\( T^{6} + \)\(13\!\cdots\!44\)\( T^{7} + \)\(10\!\cdots\!34\)\( T^{8} + \)\(13\!\cdots\!44\)\( p^{5} T^{9} + \)\(15\!\cdots\!78\)\( p^{10} T^{10} + \)\(15\!\cdots\!60\)\( p^{15} T^{11} + \)\(13\!\cdots\!36\)\( p^{20} T^{12} + 9508315264777636 p^{25} T^{13} + 60287015686 p^{30} T^{14} + 265692 p^{35} T^{15} + p^{40} T^{16} \)
97 \( 1 - 144742 T + 53143560149 T^{2} - 6952055211960972 T^{3} + \)\(13\!\cdots\!46\)\( T^{4} - \)\(15\!\cdots\!66\)\( T^{5} + \)\(20\!\cdots\!88\)\( T^{6} - \)\(20\!\cdots\!62\)\( T^{7} + \)\(21\!\cdots\!53\)\( T^{8} - \)\(20\!\cdots\!62\)\( p^{5} T^{9} + \)\(20\!\cdots\!88\)\( p^{10} T^{10} - \)\(15\!\cdots\!66\)\( p^{15} T^{11} + \)\(13\!\cdots\!46\)\( p^{20} T^{12} - 6952055211960972 p^{25} T^{13} + 53143560149 p^{30} T^{14} - 144742 p^{35} T^{15} + p^{40} T^{16} \)
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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

−4.29089110157867462122921737864, −4.15331767259955527944696463651, −3.74329212827487257045124629272, −3.69628838808457594460018572941, −3.66070991376605927114086365434, −3.60114764404426453520576059772, −3.58064796194844698581458143700, −3.41242757763010847067685850356, −3.40505452001507579778394684439, −2.80200430752710043782088438027, −2.78130757174467428107799188761, −2.69044079469129098959664048309, −2.63074838619592954925060875540, −2.53799536239801026534631347251, −2.43054357564583299476877726383, −2.41538753139170442228731857802, −2.31277437372406976350534316566, −1.71271772375828603266921365869, −1.66213004389744622088273215951, −1.62182500396203835791539049607, −1.47954114468750809948113255920, −1.37474867488948856245032918495, −0.917910839199161889213004194528, −0.874146147763291120491155032018, −0.839277795865825364333663516562, 0, 0, 0, 0, 0, 0, 0, 0, 0.839277795865825364333663516562, 0.874146147763291120491155032018, 0.917910839199161889213004194528, 1.37474867488948856245032918495, 1.47954114468750809948113255920, 1.62182500396203835791539049607, 1.66213004389744622088273215951, 1.71271772375828603266921365869, 2.31277437372406976350534316566, 2.41538753139170442228731857802, 2.43054357564583299476877726383, 2.53799536239801026534631347251, 2.63074838619592954925060875540, 2.69044079469129098959664048309, 2.78130757174467428107799188761, 2.80200430752710043782088438027, 3.40505452001507579778394684439, 3.41242757763010847067685850356, 3.58064796194844698581458143700, 3.60114764404426453520576059772, 3.66070991376605927114086365434, 3.69628838808457594460018572941, 3.74329212827487257045124629272, 4.15331767259955527944696463651, 4.29089110157867462122921737864

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

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