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Properties

Label 16-1950e8-1.1-c1e8-0-8
Degree $16$
Conductor $2.091\times 10^{26}$
Sign $1$
Analytic cond. $3.45536\times 10^{9}$
Root an. cond. $3.94598$
Motivic weight $1$
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  + 4·2-s + 6·4-s − 2·7-s + 2·9-s + 6·11-s − 6·13-s − 8·14-s − 15·16-s + 8·18-s + 6·19-s + 24·22-s − 6·23-s − 24·26-s − 12·28-s + 8·29-s − 24·32-s + 12·36-s + 10·37-s + 24·38-s + 48·43-s + 36·44-s − 24·46-s + 16·47-s + 9·49-s − 36·52-s + 32·58-s − 24·59-s + ⋯
L(s)  = 1  + 2.82·2-s + 3·4-s − 0.755·7-s + 2/3·9-s + 1.80·11-s − 1.66·13-s − 2.13·14-s − 3.75·16-s + 1.88·18-s + 1.37·19-s + 5.11·22-s − 1.25·23-s − 4.70·26-s − 2.26·28-s + 1.48·29-s − 4.24·32-s + 2·36-s + 1.64·37-s + 3.89·38-s + 7.31·43-s + 5.42·44-s − 3.53·46-s + 2.33·47-s + 9/7·49-s − 4.99·52-s + 4.20·58-s − 3.12·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(3.45536\times 10^{9}\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(20.52920170\)
\(L(\frac12)\) \(\approx\) \(20.52920170\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{4} \)
3 \( ( 1 - T^{2} + T^{4} )^{2} \)
5 \( 1 \)
13 \( 1 + 6 T + 45 T^{2} + 186 T^{3} + 848 T^{4} + 186 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
good7 \( 1 + 2 T - 5 T^{2} - 26 T^{3} - 43 T^{4} + 32 T^{5} + 338 T^{6} + 556 T^{7} - 614 T^{8} + 556 p T^{9} + 338 p^{2} T^{10} + 32 p^{3} T^{11} - 43 p^{4} T^{12} - 26 p^{5} T^{13} - 5 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \)
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 \( ( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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 \( 1 + 6 T + 59 T^{2} + 282 T^{3} + 1905 T^{4} + 10224 T^{5} + 45502 T^{6} + 270276 T^{7} + 987242 T^{8} + 270276 p T^{9} + 45502 p^{2} T^{10} + 10224 p^{3} T^{11} + 1905 p^{4} T^{12} + 282 p^{5} T^{13} + 59 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \)
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 \( 1 - 10 T - 14 T^{2} + 220 T^{3} + 1301 T^{4} - 4780 T^{5} - 35098 T^{6} + 340690 T^{7} - 2682860 T^{8} + 340690 p T^{9} - 35098 p^{2} T^{10} - 4780 p^{3} T^{11} + 1301 p^{4} T^{12} + 220 p^{5} T^{13} - 14 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \)
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 \( 1 - 48 T + 1183 T^{2} - 19920 T^{3} + 256773 T^{4} - 2696256 T^{5} + 558818 p T^{6} - 187083744 T^{7} + 1296211046 T^{8} - 187083744 p T^{9} + 558818 p^{3} T^{10} - 2696256 p^{3} T^{11} + 256773 p^{4} T^{12} - 19920 p^{5} T^{13} + 1183 p^{6} T^{14} - 48 p^{7} T^{15} + p^{8} T^{16} \)
47 \( ( 1 - 8 T + 106 T^{2} - 352 T^{3} + 4443 T^{4} - 352 p T^{5} + 106 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( 1 - 210 T^{2} + 16393 T^{4} - 540522 T^{6} + 11949540 T^{8} - 540522 p^{2} T^{10} + 16393 p^{4} T^{12} - 210 p^{6} T^{14} + p^{8} T^{16} \)
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 \( 1 - 12 T - 72 T^{2} + 1944 T^{3} - 878 T^{4} - 154212 T^{5} + 870912 T^{6} + 4319124 T^{7} - 74785245 T^{8} + 4319124 p T^{9} + 870912 p^{2} T^{10} - 154212 p^{3} T^{11} - 878 p^{4} T^{12} + 1944 p^{5} T^{13} - 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \)
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 \( ( 1 + 12 T + 168 T^{2} + 900 T^{3} + 9326 T^{4} + 900 p T^{5} + 168 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( ( 1 - 16 T + 232 T^{2} - 2048 T^{3} + 19710 T^{4} - 2048 p T^{5} + 232 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( 1 + 36 T + 528 T^{2} + 5448 T^{3} + 65458 T^{4} + 618348 T^{5} + 3080736 T^{6} + 24087660 T^{7} + 342649875 T^{8} + 24087660 p T^{9} + 3080736 p^{2} T^{10} + 618348 p^{3} T^{11} + 65458 p^{4} T^{12} + 5448 p^{5} T^{13} + 528 p^{6} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \)
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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

−4.04148011256736093464000046673, −3.75414760538866595884003854034, −3.69631661273002664909662937569, −3.65010997127378990153899376855, −3.37799697286917854497677101114, −3.30142727630169011062049277272, −3.24600730828403259924911530303, −3.05763433021768998870576842938, −2.74415414947250954553693225577, −2.70735156744703368482756098974, −2.65726511923598963794165809039, −2.65374756886063324105447613413, −2.63295046531469178458437837893, −2.22135470800740461649845895693, −2.17615795323868133415574850196, −2.13368973954189414687213192633, −1.86772957298661476980149687598, −1.53822383375999339244631131314, −1.37630045410755795727133659763, −1.15567842493576409921750497186, −1.11416443403744275024905446731, −0.973413954872906515174189438878, −0.62208951802235818498261402872, −0.57331681260560722745466899190, −0.20419612660738331341979579276, 0.20419612660738331341979579276, 0.57331681260560722745466899190, 0.62208951802235818498261402872, 0.973413954872906515174189438878, 1.11416443403744275024905446731, 1.15567842493576409921750497186, 1.37630045410755795727133659763, 1.53822383375999339244631131314, 1.86772957298661476980149687598, 2.13368973954189414687213192633, 2.17615795323868133415574850196, 2.22135470800740461649845895693, 2.63295046531469178458437837893, 2.65374756886063324105447613413, 2.65726511923598963794165809039, 2.70735156744703368482756098974, 2.74415414947250954553693225577, 3.05763433021768998870576842938, 3.24600730828403259924911530303, 3.30142727630169011062049277272, 3.37799697286917854497677101114, 3.65010997127378990153899376855, 3.69631661273002664909662937569, 3.75414760538866595884003854034, 4.04148011256736093464000046673

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

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