Properties

Label 16-3072e8-1.1-c1e8-0-2
Degree $16$
Conductor $7.932\times 10^{27}$
Sign $1$
Analytic cond. $1.31095\times 10^{11}$
Root an. cond. $4.95278$
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  − 16·7-s − 4·9-s − 32·23-s + 16·25-s + 16·31-s + 32·47-s + 104·49-s + 64·63-s − 32·71-s + 16·73-s + 48·79-s + 10·81-s + 16·89-s − 32·97-s − 48·103-s − 16·113-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 512·161-s + 163-s + 167-s + ⋯
L(s)  = 1  − 6.04·7-s − 4/3·9-s − 6.67·23-s + 16/5·25-s + 2.87·31-s + 4.66·47-s + 14.8·49-s + 8.06·63-s − 3.79·71-s + 1.87·73-s + 5.40·79-s + 10/9·81-s + 1.69·89-s − 3.24·97-s − 4.72·103-s − 1.50·113-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 + 40.3·161-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{80} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.31095\times 10^{11}\)
Root analytic conductor: \(4.95278\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3072} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{80} \cdot 3^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5636629099\)
\(L(\frac12)\) \(\approx\) \(0.5636629099\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T^{2} )^{4} \)
good5 \( 1 - 16 T^{2} + 116 T^{4} - 112 p T^{6} + 2566 T^{8} - 112 p^{3} T^{10} + 116 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \)
7 \( ( 1 + 8 T + 44 T^{2} + 24 p T^{3} + 510 T^{4} + 24 p^{2} T^{5} + 44 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( 1 - 16 T^{2} + 452 T^{4} - 7856 T^{6} + 100006 T^{8} - 7856 p^{2} T^{10} + 452 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 + 36 T^{2} - 64 T^{3} + 614 T^{4} - 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( 1 - 88 T^{2} + 3772 T^{4} - 108456 T^{6} + 2345510 T^{8} - 108456 p^{2} T^{10} + 3772 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 + 4 T + p T^{2} )^{8} \)
29 \( 1 - 144 T^{2} + 10036 T^{4} - 457904 T^{6} + 15291974 T^{8} - 457904 p^{2} T^{10} + 10036 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 - 8 T + 108 T^{2} - 616 T^{3} + 162 p T^{4} - 616 p T^{5} + 108 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( 1 - 144 T^{2} + 9092 T^{4} - 335152 T^{6} + 10897638 T^{8} - 335152 p^{2} T^{10} + 9092 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 68 T^{2} - 64 T^{3} + 3206 T^{4} - 64 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( 1 - 152 T^{2} + 13436 T^{4} - 859112 T^{6} + 42513894 T^{8} - 859112 p^{2} T^{10} + 13436 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
53 \( 1 - 208 T^{2} + 22004 T^{4} - 1673840 T^{6} + 99995782 T^{8} - 1673840 p^{2} T^{10} + 22004 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 - 152 T^{2} + 17468 T^{4} - 1286632 T^{6} + 87350758 T^{8} - 1286632 p^{2} T^{10} + 17468 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 - 208 T^{2} + 15172 T^{4} - 338160 T^{6} - 4922074 T^{8} - 338160 p^{2} T^{10} + 15172 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 88 T^{2} + 12668 T^{4} - 867240 T^{6} + 78088998 T^{8} - 867240 p^{2} T^{10} + 12668 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 16 T + 4 p T^{2} + 2640 T^{3} + 28070 T^{4} + 2640 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 8 T + 172 T^{2} - 1720 T^{3} + 16006 T^{4} - 1720 p T^{5} + 172 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 24 T + 396 T^{2} - 4344 T^{3} + 42398 T^{4} - 4344 p T^{5} + 396 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 504 T^{2} + 120700 T^{4} - 17869064 T^{6} + 1785000614 T^{8} - 17869064 p^{2} T^{10} + 120700 p^{4} T^{12} - 504 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 8 T + 188 T^{2} - 2424 T^{3} + 17894 T^{4} - 2424 p T^{5} + 188 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 16 T + 356 T^{2} + 4400 T^{3} + 49990 T^{4} + 4400 p T^{5} + 356 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−3.47058496808738343570158224918, −3.45226004029024139442581493255, −3.40406721412281381076822430234, −3.28787438005805520073519430184, −3.10387801180771243275777074766, −3.02207027965736448329444410041, −2.87171267944960950965872201056, −2.86286146525259832054211223741, −2.81155323859836854211194757412, −2.63276362975240387573131093836, −2.34561516886565198926441319357, −2.28545103328228201773939431511, −2.25362164516126077868187794983, −2.24407821694239672562934071038, −1.97843630249707181363318995380, −1.81729791590187321004881344455, −1.73051672577063483658067327866, −1.27209645076102399065263814478, −1.11005660051757577003547248165, −1.01098925357353978213060647551, −0.892732325963023267386936636844, −0.48602434227120361370740574953, −0.37688068018701853824675008288, −0.35636829192987200173181926311, −0.17498694465784485072700249497, 0.17498694465784485072700249497, 0.35636829192987200173181926311, 0.37688068018701853824675008288, 0.48602434227120361370740574953, 0.892732325963023267386936636844, 1.01098925357353978213060647551, 1.11005660051757577003547248165, 1.27209645076102399065263814478, 1.73051672577063483658067327866, 1.81729791590187321004881344455, 1.97843630249707181363318995380, 2.24407821694239672562934071038, 2.25362164516126077868187794983, 2.28545103328228201773939431511, 2.34561516886565198926441319357, 2.63276362975240387573131093836, 2.81155323859836854211194757412, 2.86286146525259832054211223741, 2.87171267944960950965872201056, 3.02207027965736448329444410041, 3.10387801180771243275777074766, 3.28787438005805520073519430184, 3.40406721412281381076822430234, 3.45226004029024139442581493255, 3.47058496808738343570158224918

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

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