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Properties

Label 16-5e24-1.1-c1e8-0-0
Degree $16$
Conductor $5.960\times 10^{16}$
Sign $1$
Analytic cond. $0.985137$
Root an. cond. $0.999064$
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  − 5·9-s − 4·11-s − 6·16-s + 10·19-s − 10·29-s + 6·31-s + 16·41-s + 50·49-s + 26·61-s − 5·64-s + 16·71-s − 30·79-s + 9·81-s + 40·89-s + 20·99-s − 24·101-s + 20·109-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 30·144-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 5/3·9-s − 1.20·11-s − 3/2·16-s + 2.29·19-s − 1.85·29-s + 1.07·31-s + 2.49·41-s + 50/7·49-s + 3.32·61-s − 5/8·64-s + 1.89·71-s − 3.37·79-s + 81-s + 4.23·89-s + 2.01·99-s − 2.38·101-s + 1.91·109-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(5^{24}\)
Sign: $1$
Analytic conductor: \(0.985137\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{24} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9430820044\)
\(L(\frac12)\) \(\approx\) \(0.9430820044\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + 3 p T^{4} + 5 T^{6} + 21 T^{8} + 5 p^{2} T^{10} + 3 p^{5} T^{12} + p^{8} T^{16} \)
3 \( 1 + 5 T^{2} + 16 T^{4} + 35 T^{6} + 31 T^{8} + 35 p^{2} T^{10} + 16 p^{4} T^{12} + 5 p^{6} T^{14} + p^{8} T^{16} \)
7 \( ( 1 - 25 T^{2} + 253 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 34 p T^{5} + 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 + 35 T^{2} + 291 T^{4} - 4775 T^{6} - 125584 T^{8} - 4775 p^{2} T^{10} + 291 p^{4} T^{12} + 35 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 + 50 T^{2} + 651 T^{4} - 13760 T^{6} - 509659 T^{8} - 13760 p^{2} T^{10} + 651 p^{4} T^{12} + 50 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 5 T + 21 T^{2} - 145 T^{3} + 956 T^{4} - 145 p T^{5} + 21 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 + 65 T^{2} + 72 p T^{4} + 32935 T^{6} + 786671 T^{8} + 32935 p^{2} T^{10} + 72 p^{5} T^{12} + 65 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 5 T - 19 T^{2} - 5 p T^{3} - 4 T^{4} - 5 p^{2} T^{5} - 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 3 T - 22 T^{2} + 159 T^{3} + 205 T^{4} + 159 p T^{5} - 22 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 + 85 T^{2} + 3096 T^{4} + 8555 T^{6} - 1746769 T^{8} + 8555 p^{2} T^{10} + 3096 p^{4} T^{12} + 85 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 - 8 T - 17 T^{2} + 254 T^{3} - 435 T^{4} + 254 p T^{5} - 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 145 T^{2} + 8853 T^{4} - 145 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( 1 + 90 T^{2} + 6351 T^{4} + 404060 T^{6} + 22389861 T^{8} + 404060 p^{2} T^{10} + 6351 p^{4} T^{12} + 90 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 65 T^{2} + 4656 T^{4} + 272695 T^{6} + 7831751 T^{8} + 272695 p^{2} T^{10} + 4656 p^{4} T^{12} + 65 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 31 T^{2} - 210 T^{3} + 2851 T^{4} - 210 p T^{5} + 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 13 T + 78 T^{2} - 941 T^{3} + 11075 T^{4} - 941 p T^{5} + 78 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 10 T^{2} + 4851 T^{4} + 94760 T^{6} + 18014741 T^{8} + 94760 p^{2} T^{10} + 4851 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 8 T - 37 T^{2} + 694 T^{3} - 2425 T^{4} + 694 p T^{5} - 37 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + 65 T^{2} - 1104 T^{4} - 418145 T^{6} - 21296209 T^{8} - 418145 p^{2} T^{10} - 1104 p^{4} T^{12} + 65 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 15 T + 21 T^{2} - 145 T^{3} + 2916 T^{4} - 145 p T^{5} + 21 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 185 T^{2} + 15096 T^{4} + 176215 T^{6} - 36076249 T^{8} + 176215 p^{2} T^{10} + 15096 p^{4} T^{12} + 185 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 20 T + 151 T^{2} - 1600 T^{3} + 21441 T^{4} - 1600 p T^{5} + 151 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 + 190 T^{2} + 23931 T^{4} + 2392340 T^{6} + 231299501 T^{8} + 2392340 p^{2} T^{10} + 23931 p^{4} T^{12} + 190 p^{6} T^{14} + p^{8} T^{16} \)
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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

−6.18292146690344596959202432049, −6.10696930285051884067001288940, −5.69746366203112538529994426349, −5.69239703882954894757895829388, −5.62349778898807181124067738562, −5.46949551689546964019650255098, −5.22098847699734075724587040962, −5.21613428887431790206537603324, −5.21327529208244492984302149512, −4.61827464846276781627458285704, −4.40696189794753581492843109110, −4.39831789353184290364788653501, −4.21038082734385381900307654475, −3.89979255863015220012740843026, −3.65572391555640403235721900256, −3.62812310243019469106802979994, −3.37012427512599979918941162197, −2.88568655688020667946843684064, −2.80177642705823829414931113637, −2.48835521061366510424077816978, −2.45431275345604760836139367433, −2.28492847914089115907774365533, −2.00941807045984137695360907115, −1.06139388462430419569641658532, −0.835682917278180444937163876946, 0.835682917278180444937163876946, 1.06139388462430419569641658532, 2.00941807045984137695360907115, 2.28492847914089115907774365533, 2.45431275345604760836139367433, 2.48835521061366510424077816978, 2.80177642705823829414931113637, 2.88568655688020667946843684064, 3.37012427512599979918941162197, 3.62812310243019469106802979994, 3.65572391555640403235721900256, 3.89979255863015220012740843026, 4.21038082734385381900307654475, 4.39831789353184290364788653501, 4.40696189794753581492843109110, 4.61827464846276781627458285704, 5.21327529208244492984302149512, 5.21613428887431790206537603324, 5.22098847699734075724587040962, 5.46949551689546964019650255098, 5.62349778898807181124067738562, 5.69239703882954894757895829388, 5.69746366203112538529994426349, 6.10696930285051884067001288940, 6.18292146690344596959202432049

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

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