Properties

Label 16-390e8-1.1-c1e8-0-4
Degree $16$
Conductor $5.352\times 10^{20}$
Sign $1$
Analytic cond. $8845.73$
Root an. cond. $1.76469$
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  − 4·3-s + 2·4-s + 6·9-s + 6·11-s − 8·12-s − 12·13-s + 16-s + 16·17-s − 6·19-s + 4·23-s − 4·25-s − 8·29-s − 24·33-s + 12·36-s + 30·37-s + 48·39-s + 14·43-s + 12·44-s − 4·48-s − 7·49-s − 64·51-s − 24·52-s + 16·53-s + 24·57-s + 24·59-s − 16·61-s − 2·64-s + ⋯
L(s)  = 1  − 2.30·3-s + 4-s + 2·9-s + 1.80·11-s − 2.30·12-s − 3.32·13-s + 1/4·16-s + 3.88·17-s − 1.37·19-s + 0.834·23-s − 4/5·25-s − 1.48·29-s − 4.17·33-s + 2·36-s + 4.93·37-s + 7.68·39-s + 2.13·43-s + 1.80·44-s − 0.577·48-s − 49-s − 8.96·51-s − 3.32·52-s + 2.19·53-s + 3.17·57-s + 3.12·59-s − 2.04·61-s − 1/4·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(8845.73\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{390} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.112194446\)
\(L(\frac12)\) \(\approx\) \(2.112194446\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{2} \)
3 \( ( 1 + T + T^{2} )^{4} \)
5 \( ( 1 + T^{2} )^{4} \)
13 \( 1 + 12 T + 45 T^{2} - 24 T^{3} - 556 T^{4} - 24 p T^{5} + 45 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
good7 \( 1 + p T^{2} - 3 T^{4} - 130 T^{6} - 1248 T^{7} - 754 T^{8} - 1248 p T^{9} - 130 p^{2} T^{10} - 3 p^{4} T^{12} + p^{7} T^{14} + p^{8} T^{16} \)
11 \( 1 - 6 T + 26 T^{2} - 84 T^{3} + 93 T^{4} - 216 T^{5} + 214 T^{6} + 366 T^{7} + 9716 T^{8} + 366 p T^{9} + 214 p^{2} T^{10} - 216 p^{3} T^{11} + 93 p^{4} T^{12} - 84 p^{5} T^{13} + 26 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \)
17 \( ( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \)
19 \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + 217 T^{4} - 2234 T^{6} - 6492 T^{7} - 40022 T^{8} - 6492 p T^{9} - 2234 p^{2} T^{10} + 217 p^{4} T^{12} + 6 p^{6} T^{13} + 31 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
23 \( 1 - 4 T - 33 T^{2} + 324 T^{3} + 113 T^{4} - 8736 T^{5} + 34370 T^{6} + 109208 T^{7} - 1157922 T^{8} + 109208 p T^{9} + 34370 p^{2} T^{10} - 8736 p^{3} T^{11} + 113 p^{4} T^{12} + 324 p^{5} T^{13} - 33 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
29 \( 1 + 8 T - 13 T^{2} - 480 T^{3} - 1615 T^{4} + 7528 T^{5} + 59362 T^{6} - 2520 T^{7} - 1294994 T^{8} - 2520 p T^{9} + 59362 p^{2} T^{10} + 7528 p^{3} T^{11} - 1615 p^{4} T^{12} - 480 p^{5} T^{13} - 13 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
31 \( 1 - 74 T^{2} + 3321 T^{4} - 90346 T^{6} + 2694452 T^{8} - 90346 p^{2} T^{10} + 3321 p^{4} T^{12} - 74 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 30 T + 514 T^{2} - 6420 T^{3} + 64101 T^{4} - 541620 T^{5} + 4034198 T^{6} - 27314130 T^{7} + 171690740 T^{8} - 27314130 p T^{9} + 4034198 p^{2} T^{10} - 541620 p^{3} T^{11} + 64101 p^{4} T^{12} - 6420 p^{5} T^{13} + 514 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \)
41 \( 1 + 80 T^{2} + 2370 T^{4} - 7488 T^{5} + 65728 T^{6} - 773760 T^{7} + 3160547 T^{8} - 773760 p T^{9} + 65728 p^{2} T^{10} - 7488 p^{3} T^{11} + 2370 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 14 T + 67 T^{2} + 134 T^{3} - 3763 T^{4} + 24232 T^{5} - 46294 T^{6} - 310492 T^{7} + 3087010 T^{8} - 310492 p T^{9} - 46294 p^{2} T^{10} + 24232 p^{3} T^{11} - 3763 p^{4} T^{12} + 134 p^{5} T^{13} + 67 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - 148 T^{2} + 14490 T^{4} - 1021616 T^{6} + 54203939 T^{8} - 1021616 p^{2} T^{10} + 14490 p^{4} T^{12} - 148 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 - 8 T + 137 T^{2} - 1268 T^{3} + 8956 T^{4} - 1268 p T^{5} + 137 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 24 T + 475 T^{2} - 6792 T^{3} + 87253 T^{4} - 932736 T^{5} + 9243862 T^{6} - 80308224 T^{7} + 654943486 T^{8} - 80308224 p T^{9} + 9243862 p^{2} T^{10} - 932736 p^{3} T^{11} + 87253 p^{4} T^{12} - 6792 p^{5} T^{13} + 475 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
61 \( ( 1 + 8 T - 62 T^{2} + 32 T^{3} + 8251 T^{4} + 32 p T^{5} - 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 24 T + 432 T^{2} - 5760 T^{3} + 69970 T^{4} - 776424 T^{5} + 7801920 T^{6} - 72545640 T^{7} + 609479859 T^{8} - 72545640 p T^{9} + 7801920 p^{2} T^{10} - 776424 p^{3} T^{11} + 69970 p^{4} T^{12} - 5760 p^{5} T^{13} + 432 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
71 \( 1 + 12 T + 128 T^{2} + 960 T^{3} + 5202 T^{4} - 11124 T^{5} - 245984 T^{6} - 4504236 T^{7} - 41845549 T^{8} - 4504236 p T^{9} - 245984 p^{2} T^{10} - 11124 p^{3} T^{11} + 5202 p^{4} T^{12} + 960 p^{5} T^{13} + 128 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \)
73 \( 1 - 192 T^{2} + 25276 T^{4} - 2537280 T^{6} + 214285254 T^{8} - 2537280 p^{2} T^{10} + 25276 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 10 T + 169 T^{2} + 1510 T^{3} + 17728 T^{4} + 1510 p T^{5} + 169 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 208 T^{2} + 27708 T^{4} - 2708144 T^{6} + 235978406 T^{8} - 2708144 p^{2} T^{10} + 27708 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 - 42 T + 1115 T^{2} - 22134 T^{3} + 362637 T^{4} - 5064156 T^{5} + 62362882 T^{6} - 684879408 T^{7} + 6796999778 T^{8} - 684879408 p T^{9} + 62362882 p^{2} T^{10} - 5064156 p^{3} T^{11} + 362637 p^{4} T^{12} - 22134 p^{5} T^{13} + 1115 p^{6} T^{14} - 42 p^{7} T^{15} + p^{8} T^{16} \)
97 \( 1 + 24 T + 408 T^{2} + 5184 T^{3} + 49906 T^{4} + 311304 T^{5} + 4896 T^{6} - 24434760 T^{7} - 344198589 T^{8} - 24434760 p T^{9} + 4896 p^{2} T^{10} + 311304 p^{3} T^{11} + 49906 p^{4} T^{12} + 5184 p^{5} T^{13} + 408 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
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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

−5.09800905159498476658462088282, −4.98215643761354715274475957794, −4.97478072078585111772673568201, −4.49396203915352315725835463499, −4.47061506942048281765745700127, −4.38847175936169729385174991285, −4.10136917398346176645609329790, −4.00644010054907671790170415197, −3.96084159815244289377413658045, −3.79991913214436395242998776301, −3.60490133179593382254443419732, −3.45196510630318440799376819684, −3.01446379869990457811133566874, −2.77713830057797031774245870276, −2.70292389482671878118778004262, −2.64847969836570994649287178487, −2.59524534362457127378917697732, −2.47768687239125723375955357895, −1.93973657953111178776505823991, −1.74699202038698265764186127350, −1.52085986021617002388305811002, −1.39277743682087190828840128906, −0.823603500455178174541225069049, −0.71739903756515032654615774137, −0.53098147218661685358093019673, 0.53098147218661685358093019673, 0.71739903756515032654615774137, 0.823603500455178174541225069049, 1.39277743682087190828840128906, 1.52085986021617002388305811002, 1.74699202038698265764186127350, 1.93973657953111178776505823991, 2.47768687239125723375955357895, 2.59524534362457127378917697732, 2.64847969836570994649287178487, 2.70292389482671878118778004262, 2.77713830057797031774245870276, 3.01446379869990457811133566874, 3.45196510630318440799376819684, 3.60490133179593382254443419732, 3.79991913214436395242998776301, 3.96084159815244289377413658045, 4.00644010054907671790170415197, 4.10136917398346176645609329790, 4.38847175936169729385174991285, 4.47061506942048281765745700127, 4.49396203915352315725835463499, 4.97478072078585111772673568201, 4.98215643761354715274475957794, 5.09800905159498476658462088282

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

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