Properties

Label 16-450e8-1.1-c1e8-0-8
Degree $16$
Conductor $1.682\times 10^{21}$
Sign $1$
Analytic cond. $27791.8$
Root an. cond. $1.89559$
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  + 2·2-s + 4-s + 4·7-s − 11-s − 13·13-s + 8·14-s + 11·17-s + 20·19-s − 2·22-s + 3·23-s + 15·25-s − 26·26-s + 4·28-s + 15·29-s − 9·31-s − 2·32-s + 22·34-s − 6·37-s + 40·38-s + 9·41-s + 12·43-s − 44-s + 6·46-s + 47-s − 22·49-s + 30·50-s − 13·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 1.51·7-s − 0.301·11-s − 3.60·13-s + 2.13·14-s + 2.66·17-s + 4.58·19-s − 0.426·22-s + 0.625·23-s + 3·25-s − 5.09·26-s + 0.755·28-s + 2.78·29-s − 1.61·31-s − 0.353·32-s + 3.77·34-s − 0.986·37-s + 6.48·38-s + 1.40·41-s + 1.82·43-s − 0.150·44-s + 0.884·46-s + 0.145·47-s − 3.14·49-s + 4.24·50-s − 1.80·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(27791.8\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(11.81006402\)
\(L(\frac12)\) \(\approx\) \(11.81006402\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
3 \( 1 \)
5 \( 1 - 3 p T^{2} + 21 p T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \)
good7 \( ( 1 - 2 T + 17 T^{2} - 30 T^{3} + 156 T^{4} - 30 p T^{5} + 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( 1 + T - 25 T^{2} - 40 T^{3} + 160 T^{4} + 653 T^{5} + 258 p T^{6} - 3940 T^{7} - 61905 T^{8} - 3940 p T^{9} + 258 p^{3} T^{10} + 653 p^{3} T^{11} + 160 p^{4} T^{12} - 40 p^{5} T^{13} - 25 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 + p T + 67 T^{2} + 164 T^{3} + 76 T^{4} - 427 T^{5} + 3190 T^{6} + 2854 p T^{7} + 173189 T^{8} + 2854 p^{2} T^{9} + 3190 p^{2} T^{10} - 427 p^{3} T^{11} + 76 p^{4} T^{12} + 164 p^{5} T^{13} + 67 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \)
17 \( 1 - 11 T + 38 T^{2} + 8 T^{3} - 474 T^{4} + 1639 T^{5} + 2235 T^{6} - 60196 T^{7} + 352544 T^{8} - 60196 p T^{9} + 2235 p^{2} T^{10} + 1639 p^{3} T^{11} - 474 p^{4} T^{12} + 8 p^{5} T^{13} + 38 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \)
19 \( ( 1 - 15 T + 93 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2}( 1 + 10 T + 41 T^{2} - 10 T^{3} - 509 T^{4} - 10 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} ) \)
23 \( 1 - 3 T + 17 T^{2} + 26 T^{3} + 866 T^{4} + 2257 T^{5} - 7690 T^{6} + 132948 T^{7} + 158519 T^{8} + 132948 p T^{9} - 7690 p^{2} T^{10} + 2257 p^{3} T^{11} + 866 p^{4} T^{12} + 26 p^{5} T^{13} + 17 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 - 15 T + 47 T^{2} + 480 T^{3} - 4732 T^{4} + 16485 T^{5} + 86 p T^{6} - 526050 T^{7} + 4240405 T^{8} - 526050 p T^{9} + 86 p^{3} T^{10} + 16485 p^{3} T^{11} - 4732 p^{4} T^{12} + 480 p^{5} T^{13} + 47 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 + 9 T + 25 T^{2} + 330 T^{3} + 2510 T^{4} + 7017 T^{5} + 86378 T^{6} + 461340 T^{7} + 675415 T^{8} + 461340 p T^{9} + 86378 p^{2} T^{10} + 7017 p^{3} T^{11} + 2510 p^{4} T^{12} + 330 p^{5} T^{13} + 25 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \)
37 \( 1 + 6 T - 27 T^{2} - 318 T^{3} - 99 T^{4} + 14106 T^{5} + 58255 T^{6} - 123234 T^{7} - 1122756 T^{8} - 123234 p T^{9} + 58255 p^{2} T^{10} + 14106 p^{3} T^{11} - 99 p^{4} T^{12} - 318 p^{5} T^{13} - 27 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 - 9 T - 55 T^{2} + 1030 T^{3} - 2470 T^{4} - 35497 T^{5} + 265658 T^{6} + 359550 T^{7} - 11328415 T^{8} + 359550 p T^{9} + 265658 p^{2} T^{10} - 35497 p^{3} T^{11} - 2470 p^{4} T^{12} + 1030 p^{5} T^{13} - 55 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
43 \( ( 1 - 6 T + 133 T^{2} - 710 T^{3} + 7916 T^{4} - 710 p T^{5} + 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - T - 77 T^{2} + 108 T^{3} + 5436 T^{4} - 2341 T^{5} - 357450 T^{6} + 29584 T^{7} + 16743019 T^{8} + 29584 p T^{9} - 357450 p^{2} T^{10} - 2341 p^{3} T^{11} + 5436 p^{4} T^{12} + 108 p^{5} T^{13} - 77 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 + 7 T + 2 T^{2} + 6 T^{3} - 794 T^{4} - 27393 T^{5} + 64385 T^{6} + 1339888 T^{7} + 6789964 T^{8} + 1339888 p T^{9} + 64385 p^{2} T^{10} - 27393 p^{3} T^{11} - 794 p^{4} T^{12} + 6 p^{5} T^{13} + 2 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 + 10 T + 142 T^{2} + 730 T^{3} + 11283 T^{4} + 22410 T^{5} + 330384 T^{6} - 1700800 T^{7} + 16244505 T^{8} - 1700800 p T^{9} + 330384 p^{2} T^{10} + 22410 p^{3} T^{11} + 11283 p^{4} T^{12} + 730 p^{5} T^{13} + 142 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
61 \( 1 - 6 T - 140 T^{2} + 210 T^{3} + 7045 T^{4} + 27242 T^{5} + 433648 T^{6} - 1657580 T^{7} - 60459295 T^{8} - 1657580 p T^{9} + 433648 p^{2} T^{10} + 27242 p^{3} T^{11} + 7045 p^{4} T^{12} + 210 p^{5} T^{13} - 140 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
67 \( 1 + 11 T + 103 T^{2} + 1672 T^{3} + 14576 T^{4} + 139271 T^{5} + 1482950 T^{6} + 11696476 T^{7} + 89866399 T^{8} + 11696476 p T^{9} + 1482950 p^{2} T^{10} + 139271 p^{3} T^{11} + 14576 p^{4} T^{12} + 1672 p^{5} T^{13} + 103 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 - 9 T + 75 T^{2} - 1220 T^{3} + 18360 T^{4} - 165937 T^{5} + 1180298 T^{6} - 14013840 T^{7} + 141940835 T^{8} - 14013840 p T^{9} + 1180298 p^{2} T^{10} - 165937 p^{3} T^{11} + 18360 p^{4} T^{12} - 1220 p^{5} T^{13} + 75 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 + 8 T + 62 T^{2} + 1114 T^{3} + 11231 T^{4} + 71208 T^{5} + 863380 T^{6} + 9235272 T^{7} + 61248289 T^{8} + 9235272 p T^{9} + 863380 p^{2} T^{10} + 71208 p^{3} T^{11} + 11231 p^{4} T^{12} + 1114 p^{5} T^{13} + 62 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
79 \( 1 + 10 T - 38 T^{2} + 320 T^{3} + 15323 T^{4} + 59210 T^{5} - 186436 T^{6} + 4386400 T^{7} + 89172925 T^{8} + 4386400 p T^{9} - 186436 p^{2} T^{10} + 59210 p^{3} T^{11} + 15323 p^{4} T^{12} + 320 p^{5} T^{13} - 38 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \)
83 \( 1 + 27 T + 237 T^{2} + 126 T^{3} - 12774 T^{4} - 113973 T^{5} - 181210 T^{6} + 11072628 T^{7} + 166729299 T^{8} + 11072628 p T^{9} - 181210 p^{2} T^{10} - 113973 p^{3} T^{11} - 12774 p^{4} T^{12} + 126 p^{5} T^{13} + 237 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 15 T - 108 T^{2} + 1640 T^{3} + 7368 T^{4} - 158465 T^{5} + 2297399 T^{6} - 297300 T^{7} - 272740720 T^{8} - 297300 p T^{9} + 2297399 p^{2} T^{10} - 158465 p^{3} T^{11} + 7368 p^{4} T^{12} + 1640 p^{5} T^{13} - 108 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \)
97 \( ( 1 + 18 T + 27 T^{2} - 1060 T^{3} - 9699 T^{4} - 1060 p T^{5} + 27 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−4.95073826756198367546128255647, −4.80854615056743770905059815945, −4.76039705733914518911560515898, −4.49860733272101886444310352838, −4.47085221419100303420137552249, −4.33888315945612470963091240551, −4.16667921516593358745001975001, −3.82108622464956884441779126716, −3.71246174956216500661705304440, −3.68082278974410415799941389105, −3.14320449872850306700899907657, −3.13541108751778977217213067803, −3.08747181847508739622640076525, −3.06099321466238628544769289602, −2.79664932788655801474686401926, −2.74390655219393818171254703756, −2.55714357764392171769783156931, −2.50370634293060288506765605087, −1.95368324158015536629900332659, −1.57351132688070417888829746402, −1.46639640858584809221119684048, −1.39522249770388458076991751870, −1.27187375771751555041006509621, −0.810720701918057698259400249482, −0.49629825910657292722659924570, 0.49629825910657292722659924570, 0.810720701918057698259400249482, 1.27187375771751555041006509621, 1.39522249770388458076991751870, 1.46639640858584809221119684048, 1.57351132688070417888829746402, 1.95368324158015536629900332659, 2.50370634293060288506765605087, 2.55714357764392171769783156931, 2.74390655219393818171254703756, 2.79664932788655801474686401926, 3.06099321466238628544769289602, 3.08747181847508739622640076525, 3.13541108751778977217213067803, 3.14320449872850306700899907657, 3.68082278974410415799941389105, 3.71246174956216500661705304440, 3.82108622464956884441779126716, 4.16667921516593358745001975001, 4.33888315945612470963091240551, 4.47085221419100303420137552249, 4.49860733272101886444310352838, 4.76039705733914518911560515898, 4.80854615056743770905059815945, 4.95073826756198367546128255647

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

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