Properties

Label 16-891e8-1.1-c1e8-0-2
Degree $16$
Conductor $3.972\times 10^{23}$
Sign $1$
Analytic cond. $6.56505\times 10^{6}$
Root an. cond. $2.66733$
Motivic weight $1$
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  + 3·2-s + 2·4-s + 5-s + 3·7-s − 9·8-s + 3·10-s − 9·11-s + 9·13-s + 9·14-s − 26·16-s + 4·17-s − 20·19-s + 2·20-s − 27·22-s + 4·23-s + 25-s + 27·26-s + 6·28-s + 10·29-s − 8·31-s − 18·32-s + 12·34-s + 3·35-s − 6·37-s − 60·38-s − 9·40-s − 23·41-s + ⋯
L(s)  = 1  + 2.12·2-s + 4-s + 0.447·5-s + 1.13·7-s − 3.18·8-s + 0.948·10-s − 2.71·11-s + 2.49·13-s + 2.40·14-s − 6.5·16-s + 0.970·17-s − 4.58·19-s + 0.447·20-s − 5.75·22-s + 0.834·23-s + 1/5·25-s + 5.29·26-s + 1.13·28-s + 1.85·29-s − 1.43·31-s − 3.18·32-s + 2.05·34-s + 0.507·35-s − 0.986·37-s − 9.73·38-s − 1.42·40-s − 3.59·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{32} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(6.56505\times 10^{6}\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{32} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6179816070\)
\(L(\frac12)\) \(\approx\) \(0.6179816070\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + 9 T + 40 T^{2} + 171 T^{3} + 669 T^{4} + 171 p T^{5} + 40 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
good2 \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + 3 T^{4} + 3 p T^{5} + 7 p T^{6} - 3 p^{4} T^{7} + 113 T^{8} - 3 p^{5} T^{9} + 7 p^{3} T^{10} + 3 p^{4} T^{11} + 3 p^{4} T^{12} - 3 p^{6} T^{13} + 7 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
5 \( 1 - T + 21 T^{3} - 46 T^{4} + 6 T^{5} + 16 p T^{6} - 436 T^{7} + 291 T^{8} - 436 p T^{9} + 16 p^{3} T^{10} + 6 p^{3} T^{11} - 46 p^{4} T^{12} + 21 p^{5} T^{13} - p^{7} T^{15} + p^{8} T^{16} \)
7 \( 1 - 3 T + p T^{2} - 36 T^{3} + 108 T^{4} - 219 T^{5} + 122 p T^{6} - 2628 T^{7} + 5483 T^{8} - 2628 p T^{9} + 122 p^{3} T^{10} - 219 p^{3} T^{11} + 108 p^{4} T^{12} - 36 p^{5} T^{13} + p^{7} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 - 9 T + 63 T^{2} - 392 T^{3} + 2148 T^{4} - 10637 T^{5} + 46966 T^{6} - 191706 T^{7} + 723193 T^{8} - 191706 p T^{9} + 46966 p^{2} T^{10} - 10637 p^{3} T^{11} + 2148 p^{4} T^{12} - 392 p^{5} T^{13} + 63 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
17 \( ( 1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 20 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 70 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 2 T - p T^{2} + 38 T^{3} + 108 T^{4} + 38 p T^{5} - p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 10 T + 69 T^{2} + 90 T^{3} - 2860 T^{4} + 24420 T^{5} - 22309 T^{6} - 509200 T^{7} + 5518359 T^{8} - 509200 p T^{9} - 22309 p^{2} T^{10} + 24420 p^{3} T^{11} - 2860 p^{4} T^{12} + 90 p^{5} T^{13} + 69 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 + 8 T + 61 T^{2} - 68 T^{3} - 1534 T^{4} - 556 p T^{5} - 7051 T^{6} + 306886 T^{7} + 4196107 T^{8} + 306886 p T^{9} - 7051 p^{2} T^{10} - 556 p^{4} T^{11} - 1534 p^{4} T^{12} - 68 p^{5} T^{13} + 61 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
37 \( ( 1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 155 p T^{5} - 18 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 + 23 T + 321 T^{2} + 2862 T^{3} + 17186 T^{4} + 56439 T^{5} - 110206 T^{6} - 2943754 T^{7} - 24697863 T^{8} - 2943754 p T^{9} - 110206 p^{2} T^{10} + 56439 p^{3} T^{11} + 17186 p^{4} T^{12} + 2862 p^{5} T^{13} + 321 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \)
43 \( ( 1 + 8 T - 33 T^{2} + 88 T^{3} + 4808 T^{4} + 88 p T^{5} - 33 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 3 T + 52 T^{2} + 39 T^{3} - 132 T^{4} + 12876 T^{5} - 1928 p T^{6} + 250602 T^{7} - 2713027 T^{8} + 250602 p T^{9} - 1928 p^{3} T^{10} + 12876 p^{3} T^{11} - 132 p^{4} T^{12} + 39 p^{5} T^{13} + 52 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
53 \( ( 1 - 6 T + 23 T^{2} + 120 T^{3} - 1319 T^{4} + 120 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 20 T + 269 T^{2} - 1560 T^{3} + 2710 T^{4} + 53340 T^{5} + 63041 T^{6} - 7076090 T^{7} + 90705499 T^{8} - 7076090 p T^{9} + 63041 p^{2} T^{10} + 53340 p^{3} T^{11} + 2710 p^{4} T^{12} - 1560 p^{5} T^{13} + 269 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 + 3 T + 16 T^{2} - 903 T^{3} - 6174 T^{4} - 5586 T^{5} + 95984 T^{6} + 3140316 T^{7} + 2204147 T^{8} + 3140316 p T^{9} + 95984 p^{2} T^{10} - 5586 p^{3} T^{11} - 6174 p^{4} T^{12} - 903 p^{5} T^{13} + 16 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \)
67 \( ( 1 + T - 32 T^{2} - 101 T^{3} - 3467 T^{4} - 101 p T^{5} - 32 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 27 T + 253 T^{2} + 819 T^{3} + 100 T^{4} + 819 p T^{5} + 253 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 6 T - 57 T^{2} + 130 T^{3} + 4761 T^{4} + 130 p T^{5} - 57 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 + 5 T + 19 T^{2} - 1400 T^{3} - 13360 T^{4} - 37435 T^{5} + 244766 T^{6} + 8996620 T^{7} + 41542459 T^{8} + 8996620 p T^{9} + 244766 p^{2} T^{10} - 37435 p^{3} T^{11} - 13360 p^{4} T^{12} - 1400 p^{5} T^{13} + 19 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 + 21 T + 353 T^{2} + 3678 T^{3} + 40158 T^{4} + 397893 T^{5} + 4806046 T^{6} + 50181804 T^{7} + 508233323 T^{8} + 50181804 p T^{9} + 4806046 p^{2} T^{10} + 397893 p^{3} T^{11} + 40158 p^{4} T^{12} + 3678 p^{5} T^{13} + 353 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \)
89 \( ( 1 - 10 T + 183 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \)
97 \( 1 - 33 T + 552 T^{2} - 4411 T^{3} - 2442 T^{4} + 479666 T^{5} - 5140616 T^{6} + 22640112 T^{7} - 41156417 T^{8} + 22640112 p T^{9} - 5140616 p^{2} T^{10} + 479666 p^{3} T^{11} - 2442 p^{4} T^{12} - 4411 p^{5} T^{13} + 552 p^{6} T^{14} - 33 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.38901286378811397079472541618, −4.16986916589665293680428434535, −4.08834279090646556402429670619, −4.05639466576047948495721223216, −3.93722384493155457676010991664, −3.87752854261540531062013616525, −3.41456743951684238719132750269, −3.41037307289947161918938431501, −3.33948699796223001229033983555, −3.11717240171267703815137390953, −2.97334185865630287011434867998, −2.95368857069207500490553582026, −2.93179822426877615526974332875, −2.80322057047166928682795845695, −2.21783501077770317595399763054, −2.14806709587009655147082975173, −2.12569265216218013868817586564, −2.06901913463720509589744936467, −1.79046294989799321267323050959, −1.70679342235085118965729824256, −1.39094185801715938136664262075, −1.10180021620944832357681118938, −0.75696715580205815787365810163, −0.39856015733096538977455274397, −0.099245298131578317865140551183, 0.099245298131578317865140551183, 0.39856015733096538977455274397, 0.75696715580205815787365810163, 1.10180021620944832357681118938, 1.39094185801715938136664262075, 1.70679342235085118965729824256, 1.79046294989799321267323050959, 2.06901913463720509589744936467, 2.12569265216218013868817586564, 2.14806709587009655147082975173, 2.21783501077770317595399763054, 2.80322057047166928682795845695, 2.93179822426877615526974332875, 2.95368857069207500490553582026, 2.97334185865630287011434867998, 3.11717240171267703815137390953, 3.33948699796223001229033983555, 3.41037307289947161918938431501, 3.41456743951684238719132750269, 3.87752854261540531062013616525, 3.93722384493155457676010991664, 4.05639466576047948495721223216, 4.08834279090646556402429670619, 4.16986916589665293680428434535, 4.38901286378811397079472541618

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

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