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Properties

Label 16-1800e8-1.1-c1e8-0-1
Degree $16$
Conductor $1.102\times 10^{26}$
Sign $1$
Analytic cond. $1.82136\times 10^{9}$
Root an. cond. $3.79118$
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 + 4·4-s + 8·7-s + 4·8-s + 16·14-s + 6·16-s − 8·23-s + 32·28-s + 8·31-s − 16·46-s + 4·49-s + 32·56-s + 16·62-s − 8·64-s + 40·71-s − 16·73-s − 16·79-s − 32·92-s − 8·97-s + 8·98-s + 64·103-s + 48·112-s + 32·113-s + 56·121-s + 32·124-s + 127-s − 32·128-s + ⋯
L(s)  = 1  + 1.41·2-s + 2·4-s + 3.02·7-s + 1.41·8-s + 4.27·14-s + 3/2·16-s − 1.66·23-s + 6.04·28-s + 1.43·31-s − 2.35·46-s + 4/7·49-s + 4.27·56-s + 2.03·62-s − 64-s + 4.74·71-s − 1.87·73-s − 1.80·79-s − 3.33·92-s − 0.812·97-s + 0.808·98-s + 6.30·103-s + 4.53·112-s + 3.01·113-s + 5.09·121-s + 2.87·124-s + 0.0887·127-s − 2.82·128-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{16} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1.82136\times 10^{9}\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.170146269\)
\(L(\frac12)\) \(\approx\) \(9.170146269\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T + p^{2} T^{3} - 3 p T^{4} + p^{3} T^{5} - p^{4} T^{7} + p^{4} T^{8} \)
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

−3.88266718844882848367706692076, −3.78175245245725260656146012182, −3.60319906397029634411651089290, −3.58623771710889528355365417189, −3.37916096001324588004010374968, −3.26004420680977716841618284171, −3.13527494556327164187592507861, −3.01610735754576269658219980119, −3.01180060900408815649492181154, −2.99546464555679630585753658182, −2.62937129410050142870613297897, −2.33045672107811287146322421657, −2.17670013114108236161866432605, −2.15790180882278036575941617438, −2.12392690817570772892093802394, −2.05985667270012139230754413753, −1.74945761772444841714631612546, −1.64884348801313836695388434084, −1.63017324079792585932430427446, −1.51336513844007898995466603379, −1.10145356073432137706598020735, −0.884012783868078361687189348819, −0.73577810483741936086093001696, −0.72468132214361402131071724136, −0.11656170130339695861720440271, 0.11656170130339695861720440271, 0.72468132214361402131071724136, 0.73577810483741936086093001696, 0.884012783868078361687189348819, 1.10145356073432137706598020735, 1.51336513844007898995466603379, 1.63017324079792585932430427446, 1.64884348801313836695388434084, 1.74945761772444841714631612546, 2.05985667270012139230754413753, 2.12392690817570772892093802394, 2.15790180882278036575941617438, 2.17670013114108236161866432605, 2.33045672107811287146322421657, 2.62937129410050142870613297897, 2.99546464555679630585753658182, 3.01180060900408815649492181154, 3.01610735754576269658219980119, 3.13527494556327164187592507861, 3.26004420680977716841618284171, 3.37916096001324588004010374968, 3.58623771710889528355365417189, 3.60319906397029634411651089290, 3.78175245245725260656146012182, 3.88266718844882848367706692076

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

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