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Properties

Label 16-20e24-1.1-c1e8-0-1
Degree $16$
Conductor $1.678\times 10^{31}$
Sign $1$
Analytic cond. $2.77291\times 10^{14}$
Root an. cond. $7.99251$
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  − 8·9-s + 8·13-s − 20·17-s + 16·29-s + 28·37-s + 8·41-s − 20·49-s + 48·53-s + 28·61-s + 31·81-s + 20·89-s − 16·97-s + 16·101-s + 28·109-s − 64·113-s − 64·117-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 160·153-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 8/3·9-s + 2.21·13-s − 4.85·17-s + 2.97·29-s + 4.60·37-s + 1.24·41-s − 2.85·49-s + 6.59·53-s + 3.58·61-s + 31/9·81-s + 2.11·89-s − 1.62·97-s + 1.59·101-s + 2.68·109-s − 6.02·113-s − 5.91·117-s − 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 12.9·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 5^{24}\)
Sign: $1$
Analytic conductor: \(2.77291\times 10^{14}\)
Root analytic conductor: \(7.99251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 5^{24} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.700005342\)
\(L(\frac12)\) \(\approx\) \(4.700005342\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 8 T^{2} + 11 p T^{4} + 140 T^{6} + 521 T^{8} + 140 p^{2} T^{10} + 11 p^{5} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 + 20 T^{2} + 249 T^{4} + 300 p T^{6} + 15721 T^{8} + 300 p^{3} T^{10} + 249 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 + 24 T^{2} + 508 T^{4} + 8232 T^{6} + 94054 T^{8} + 8232 p^{2} T^{10} + 508 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 - 4 T + 20 T^{2} - 84 T^{3} + 326 T^{4} - 84 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 10 T + 60 T^{2} + 270 T^{3} + 1158 T^{4} + 270 p T^{5} + 60 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 + 12 T^{2} + 868 T^{4} + 8724 T^{6} + 408790 T^{8} + 8724 p^{2} T^{10} + 868 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 + 68 T^{2} + 1993 T^{4} + 54020 T^{6} + 1471361 T^{8} + 54020 p^{2} T^{10} + 1993 p^{4} T^{12} + 68 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 - 8 T + 97 T^{2} - 556 T^{3} + 3969 T^{4} - 556 p T^{5} + 97 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 + 24 T^{2} + 3548 T^{4} + 61032 T^{6} + 4945734 T^{8} + 61032 p^{2} T^{10} + 3548 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \)
37 \( ( 1 - 14 T + 192 T^{2} - 1514 T^{3} + 11390 T^{4} - 1514 p T^{5} + 192 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - 4 T + 73 T^{2} + 68 T^{3} + 1929 T^{4} + 68 p T^{5} + 73 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 + 240 T^{2} + 28049 T^{4} + 2073900 T^{6} + 105966081 T^{8} + 2073900 p^{2} T^{10} + 28049 p^{4} T^{12} + 240 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 + 200 T^{2} + 19489 T^{4} + 1261940 T^{6} + 64386401 T^{8} + 1261940 p^{2} T^{10} + 19489 p^{4} T^{12} + 200 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \)
59 \( 1 + 136 T^{2} + 11868 T^{4} + 862008 T^{6} + 50721574 T^{8} + 862008 p^{2} T^{10} + 11868 p^{4} T^{12} + 136 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 - 14 T + 143 T^{2} - 1212 T^{3} + 9649 T^{4} - 1212 p T^{5} + 143 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 10604 T^{4} - 49920 T^{6} + 63631046 T^{8} - 49920 p^{2} T^{10} + 10604 p^{4} T^{12} + p^{8} T^{16} \)
71 \( 1 + 204 T^{2} + 27748 T^{4} + 40812 p T^{6} + 229067254 T^{8} + 40812 p^{3} T^{10} + 27748 p^{4} T^{12} + 204 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 + 100 T^{2} + 1320 T^{3} + 1558 T^{4} + 1320 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( 1 + 516 T^{2} + 123428 T^{4} + 17884668 T^{6} + 1716066614 T^{8} + 17884668 p^{2} T^{10} + 123428 p^{4} T^{12} + 516 p^{6} T^{14} + p^{8} T^{16} \)
83 \( 1 + 308 T^{2} + 54433 T^{4} + 6623540 T^{6} + 622587041 T^{8} + 6623540 p^{2} T^{10} + 54433 p^{4} T^{12} + 308 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 5 T + 173 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \)
97 \( ( 1 + 8 T + 212 T^{2} + 2160 T^{3} + 26326 T^{4} + 2160 p T^{5} + 212 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−3.09806446577863394157265177351, −2.85978578980812218921895046936, −2.83991762736390914218615055835, −2.70831345339333939905130337695, −2.63985297716866120404912955312, −2.62641569552021543959257403524, −2.58995743024813824709380012432, −2.43895611447110990553162336278, −2.42500412737178918327211903778, −2.31940153245486958530738604257, −2.16208632742424021680888116639, −2.03206129288583951208756876839, −1.93304456447051762402947849410, −1.81074689371538607339967538510, −1.52623686029442503304823221339, −1.38384489955975925309315536181, −1.34738195943285947861613731939, −1.05924516928649688906257372190, −1.03333785088558437204358132046, −0.912360102410202237244780234330, −0.71289647179829018924797423522, −0.65006802854031472995453900354, −0.41340887068424866068307846404, −0.40411924561238761902244255434, −0.12909515801113239796737395218, 0.12909515801113239796737395218, 0.40411924561238761902244255434, 0.41340887068424866068307846404, 0.65006802854031472995453900354, 0.71289647179829018924797423522, 0.912360102410202237244780234330, 1.03333785088558437204358132046, 1.05924516928649688906257372190, 1.34738195943285947861613731939, 1.38384489955975925309315536181, 1.52623686029442503304823221339, 1.81074689371538607339967538510, 1.93304456447051762402947849410, 2.03206129288583951208756876839, 2.16208632742424021680888116639, 2.31940153245486958530738604257, 2.42500412737178918327211903778, 2.43895611447110990553162336278, 2.58995743024813824709380012432, 2.62641569552021543959257403524, 2.63985297716866120404912955312, 2.70831345339333939905130337695, 2.83991762736390914218615055835, 2.85978578980812218921895046936, 3.09806446577863394157265177351

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

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