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Properties

Label 16-7200e8-1.1-c1e8-0-8
Degree $16$
Conductor $7.222\times 10^{30}$
Sign $1$
Analytic cond. $1.19364\times 10^{14}$
Root an. cond. $7.58236$
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  − 8·7-s + 8·23-s − 8·31-s + 4·49-s − 40·71-s − 16·73-s + 16·79-s − 8·97-s − 64·103-s + 32·113-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 64·161-s + 163-s + 167-s + 60·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 3.02·7-s + 1.66·23-s − 1.43·31-s + 4/7·49-s − 4.74·71-s − 1.87·73-s + 1.80·79-s − 0.812·97-s − 6.30·103-s + 3.01·113-s + 5.09·121-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 + 0.0798·157-s − 5.04·161-s + 0.0783·163-s + 0.0773·167-s + 4.61·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{16} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1.19364\times 10^{14}\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.530684578\)
\(L(\frac12)\) \(\approx\) \(2.530684578\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( ( 1 + 4 T + 22 T^{2} + 72 T^{3} + 211 T^{4} + 72 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( 1 - 60 T^{2} + 1802 T^{4} - 36176 T^{6} + 538099 T^{8} - 36176 p^{2} T^{10} + 1802 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 28 T^{2} - 104 T^{3} + 350 T^{4} - 104 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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 \( ( 1 - 4 T + 36 T^{2} - 124 T^{3} + 510 T^{4} - 124 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( 1 - 168 T^{2} + 14140 T^{4} - 808664 T^{6} + 34400998 T^{8} - 808664 p^{2} T^{10} + 14140 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
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 \( 1 - 100 T^{2} + 7434 T^{4} - 377968 T^{6} + 18442035 T^{8} - 377968 p^{2} T^{10} + 7434 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 116 T^{2} - 256 T^{3} + 6310 T^{4} - 256 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( 1 - 168 T^{2} + 16780 T^{4} - 1266264 T^{6} + 74218758 T^{8} - 1266264 p^{2} T^{10} + 16780 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \)
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 \( 1 - 164 T^{2} + 20746 T^{4} - 1788592 T^{6} + 137741171 T^{8} - 1788592 p^{2} T^{10} + 20746 p^{4} T^{12} - 164 p^{6} T^{14} + p^{8} T^{16} \)
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 \( ( 1 + 8 T + 124 T^{2} + 888 T^{3} + 7014 T^{4} + 888 p T^{5} + 124 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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 \( 1 - 296 T^{2} + 53884 T^{4} - 6923736 T^{6} + 654380710 T^{8} - 6923736 p^{2} T^{10} + 53884 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} \)
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 \( ( 1 + 4 T + 186 T^{2} + 816 T^{3} + 26147 T^{4} + 816 p T^{5} + 186 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−3.07699804230215937638578903956, −3.05506051421334354308901473408, −3.05481923262183232479697529129, −3.02852084018506243094877463724, −2.81374142585470918815982416314, −2.74103741588037917704770001539, −2.71835642037563760169750496287, −2.53078845195904581919247439215, −2.51794470497285361080117078930, −2.05200140953492433358045762877, −1.97600188551686354053529862991, −1.96732009032546972029252267859, −1.93490272302299396812333381828, −1.75110696264098023588771673303, −1.61500319025012331921068642137, −1.55128918278656377903905981653, −1.51839862921497071923731314970, −1.13443984574856060602858209684, −0.978347558055241364444414408545, −0.946521519707470345132735664813, −0.76473016138312442922052131849, −0.54141323676718925279763665938, −0.31901497607796968337153564497, −0.27629138960160382909090392576, −0.20851816740616069387154272685, 0.20851816740616069387154272685, 0.27629138960160382909090392576, 0.31901497607796968337153564497, 0.54141323676718925279763665938, 0.76473016138312442922052131849, 0.946521519707470345132735664813, 0.978347558055241364444414408545, 1.13443984574856060602858209684, 1.51839862921497071923731314970, 1.55128918278656377903905981653, 1.61500319025012331921068642137, 1.75110696264098023588771673303, 1.93490272302299396812333381828, 1.96732009032546972029252267859, 1.97600188551686354053529862991, 2.05200140953492433358045762877, 2.51794470497285361080117078930, 2.53078845195904581919247439215, 2.71835642037563760169750496287, 2.74103741588037917704770001539, 2.81374142585470918815982416314, 3.02852084018506243094877463724, 3.05481923262183232479697529129, 3.05506051421334354308901473408, 3.07699804230215937638578903956

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

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