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Properties

Label 16-600e8-1.1-c1e8-0-8
Degree $16$
Conductor $1.680\times 10^{22}$
Sign $1$
Analytic cond. $277604.$
Root an. cond. $2.18884$
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 + 8·3-s + 16·6-s + 36·9-s + 6·16-s + 72·18-s + 120·27-s + 8·31-s + 8·32-s + 8·43-s + 48·48-s + 28·49-s − 8·53-s + 240·54-s + 16·62-s + 8·64-s − 24·67-s − 40·71-s + 16·79-s + 330·81-s + 32·83-s + 16·86-s + 64·93-s + 64·96-s + 56·98-s − 16·106-s + 32·107-s + ⋯
L(s)  = 1  + 1.41·2-s + 4.61·3-s + 6.53·6-s + 12·9-s + 3/2·16-s + 16.9·18-s + 23.0·27-s + 1.43·31-s + 1.41·32-s + 1.21·43-s + 6.92·48-s + 4·49-s − 1.09·53-s + 32.6·54-s + 2.03·62-s + 64-s − 2.93·67-s − 4.74·71-s + 1.80·79-s + 36.6·81-s + 3.51·83-s + 1.72·86-s + 6.63·93-s + 6.53·96-s + 5.65·98-s − 1.55·106-s + 3.09·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(277604.\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(117.1037991\)
\(L(\frac12)\) \(\approx\) \(117.1037991\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T + p^{2} T^{2} - p^{3} T^{3} + 5 p T^{4} - p^{4} T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
3 \( ( 1 - T )^{8} \)
5 \( 1 \)
good7 \( 1 - 4 p T^{2} + 330 T^{4} - 2224 T^{6} + 13203 T^{8} - 2224 p^{2} T^{10} + 330 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
11 \( 1 - 56 T^{2} + 1612 T^{4} - 29896 T^{6} + 388998 T^{8} - 29896 p^{2} T^{10} + 1612 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 + 30 T^{2} - 32 T^{3} + 451 T^{4} - 32 p T^{5} + 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( 1 - 56 T^{2} + 1484 T^{4} - 24968 T^{6} + 374950 T^{8} - 24968 p^{2} T^{10} + 1484 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
19 \( 1 - 36 T^{2} + 1546 T^{4} - 35120 T^{6} + 832243 T^{8} - 35120 p^{2} T^{10} + 1546 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 - 56 T^{2} + 1324 T^{4} - 1592 p T^{6} + 1094310 T^{8} - 1592 p^{3} T^{10} + 1324 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
29 \( 1 - 88 T^{2} + 4780 T^{4} - 171048 T^{6} + 5385990 T^{8} - 171048 p^{2} T^{10} + 4780 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 - 4 T + 54 T^{2} - 168 T^{3} + 2099 T^{4} - 168 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 84 T^{2} - 128 T^{3} + 3542 T^{4} - 128 p T^{5} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 100 T^{2} + 56 T^{3} + 126 p T^{4} + 56 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 4 T + 58 T^{2} - 336 T^{3} + 3379 T^{4} - 336 p T^{5} + 58 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 232 T^{2} + 26076 T^{4} - 1910872 T^{6} + 102863686 T^{8} - 1910872 p^{2} T^{10} + 26076 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 + 4 T + 92 T^{2} - 44 T^{3} + 3982 T^{4} - 44 p T^{5} + 92 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 40 T^{2} + 2364 T^{4} - 112984 T^{6} + 20250598 T^{8} - 112984 p^{2} T^{10} + 2364 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 - 252 T^{2} + 32650 T^{4} - 2942672 T^{6} + 202734451 T^{8} - 2942672 p^{2} T^{10} + 32650 p^{4} T^{12} - 252 p^{6} T^{14} + p^{8} T^{16} \)
67 \( ( 1 + 12 T + 154 T^{2} + 1520 T^{3} + 16755 T^{4} + 1520 p T^{5} + 154 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 20 T + 380 T^{2} + 4188 T^{3} + 43342 T^{4} + 4188 p T^{5} + 380 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 184 T^{2} + 15196 T^{4} - 1235336 T^{6} + 104948486 T^{8} - 1235336 p^{2} T^{10} + 15196 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 8 T + 132 T^{2} - 1032 T^{3} + 16454 T^{4} - 1032 p T^{5} + 132 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( ( 1 - 16 T + 276 T^{2} - 2576 T^{3} + 30070 T^{4} - 2576 p T^{5} + 276 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( ( 1 + 132 T^{2} - 64 T^{3} + 18534 T^{4} - 64 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( 1 - 356 T^{2} + 80362 T^{4} - 11927760 T^{6} + 1353370099 T^{8} - 11927760 p^{2} T^{10} + 80362 p^{4} T^{12} - 356 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.47030542909373964019411101685, −4.44599329893579208794227444084, −4.42267411532261301760436902602, −4.35262593059949547487455998694, −4.23634401932990061232852529177, −3.76862831431111336616735592613, −3.65401110249242816254299342341, −3.58095097182615840034577816759, −3.48828450908188872980928251306, −3.46458769370075850693271056108, −3.44025775314523136008444902405, −3.10983942703431154316713992371, −2.86085569548846092714062913626, −2.79445272217427513918392674720, −2.55970773399784185044009798416, −2.51770187819489101504788724970, −2.47916611851055270062720256990, −2.16118198385817224865390520683, −2.03310905206904008155136504438, −1.74808743734240588696126617749, −1.62156460732819543500761250582, −1.51253127233738562753896060535, −1.03607698790337844962063647822, −0.898075358576082711519951453598, −0.71989644200762941532033453472, 0.71989644200762941532033453472, 0.898075358576082711519951453598, 1.03607698790337844962063647822, 1.51253127233738562753896060535, 1.62156460732819543500761250582, 1.74808743734240588696126617749, 2.03310905206904008155136504438, 2.16118198385817224865390520683, 2.47916611851055270062720256990, 2.51770187819489101504788724970, 2.55970773399784185044009798416, 2.79445272217427513918392674720, 2.86085569548846092714062913626, 3.10983942703431154316713992371, 3.44025775314523136008444902405, 3.46458769370075850693271056108, 3.48828450908188872980928251306, 3.58095097182615840034577816759, 3.65401110249242816254299342341, 3.76862831431111336616735592613, 4.23634401932990061232852529177, 4.35262593059949547487455998694, 4.42267411532261301760436902602, 4.44599329893579208794227444084, 4.47030542909373964019411101685

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

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