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Properties

Label 16-672e8-1.1-c1e8-0-10
Degree $16$
Conductor $4.159\times 10^{22}$
Sign $1$
Analytic cond. $687339.$
Root an. cond. $2.31645$
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  + 8·3-s + 4·7-s + 36·9-s + 8·19-s + 32·21-s + 12·25-s + 120·27-s − 16·31-s + 8·37-s + 16·47-s + 8·49-s + 16·53-s + 64·57-s + 32·59-s + 144·63-s + 96·75-s + 330·81-s + 16·83-s − 128·93-s − 48·103-s + 64·111-s + 20·121-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + ⋯
L(s)  = 1  + 4.61·3-s + 1.51·7-s + 12·9-s + 1.83·19-s + 6.98·21-s + 12/5·25-s + 23.0·27-s − 2.87·31-s + 1.31·37-s + 2.33·47-s + 8/7·49-s + 2.19·53-s + 8.47·57-s + 4.16·59-s + 18.1·63-s + 11.0·75-s + 36.6·81-s + 1.75·83-s − 13.2·93-s − 4.72·103-s + 6.07·111-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(687339.\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(104.5259132\)
\(L(\frac12)\) \(\approx\) \(104.5259132\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T )^{8} \)
7 \( 1 - 4 T + 8 T^{2} - 20 T^{3} + 46 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
good5 \( 1 - 12 T^{2} + 56 T^{4} + 28 T^{6} - 1074 T^{8} + 28 p^{2} T^{10} + 56 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 20 T^{2} + 296 T^{4} - 2268 T^{6} + 20718 T^{8} - 2268 p^{2} T^{10} + 296 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \)
17 \( 1 - 108 T^{2} + 5432 T^{4} - 166628 T^{6} + 3420078 T^{8} - 166628 p^{2} T^{10} + 5432 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 - 4 T + 32 T^{2} - 132 T^{3} + 558 T^{4} - 132 p T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 100 T^{2} + 5000 T^{4} - 174764 T^{6} + 4622926 T^{8} - 174764 p^{2} T^{10} + 5000 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 36 T^{2} + 96 T^{3} + 390 T^{4} + 96 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 8 T + 100 T^{2} + 616 T^{3} + 4534 T^{4} + 616 p T^{5} + 100 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 4 T + 40 T^{2} + 292 T^{3} - 866 T^{4} + 292 p T^{5} + 40 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 204 T^{2} + 21176 T^{4} - 1439172 T^{6} + 69360622 T^{8} - 1439172 p^{2} T^{10} + 21176 p^{4} T^{12} - 204 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 192 T^{2} + 18396 T^{4} - 1162560 T^{6} + 55940774 T^{8} - 1162560 p^{2} T^{10} + 18396 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 - 8 T + 108 T^{2} - 872 T^{3} + 6758 T^{4} - 872 p T^{5} + 108 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 8 T + 132 T^{2} - 632 T^{3} + 7718 T^{4} - 632 p T^{5} + 132 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
61 \( 1 - 184 T^{2} + 16284 T^{4} - 814280 T^{6} + 38728934 T^{8} - 814280 p^{2} T^{10} + 16284 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 240 T^{2} + 37436 T^{4} - 3815056 T^{6} + 299829030 T^{8} - 3815056 p^{2} T^{10} + 37436 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 - 484 T^{2} + 107912 T^{4} - 14382508 T^{6} + 1250123214 T^{8} - 14382508 p^{2} T^{10} + 107912 p^{4} T^{12} - 484 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 408 T^{2} + 80380 T^{4} - 10037032 T^{6} + 869620166 T^{8} - 10037032 p^{2} T^{10} + 80380 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 - 368 T^{2} + 73436 T^{4} - 9593744 T^{6} + 892827078 T^{8} - 9593744 p^{2} T^{10} + 73436 p^{4} T^{12} - 368 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 - 8 T + 124 T^{2} - 1480 T^{3} + 7318 T^{4} - 1480 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 + 52 T^{2} - 328 T^{4} + 280764 T^{6} + 120770670 T^{8} + 280764 p^{2} T^{10} - 328 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 - 344 T^{2} + 58172 T^{4} - 6404840 T^{6} + 615495942 T^{8} - 6404840 p^{2} T^{10} + 58172 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.52724933738373377326960140233, −4.18785379853488530109142632889, −4.16609295625747102122955072515, −3.98402774684297691284390846315, −3.90313934560955782230892078976, −3.87711647648630139783688324353, −3.71599298018287165749167569589, −3.59966196725121410229560625292, −3.59624359834717057980880729075, −3.09006779093623381268103510236, −3.03320621859166994473393779587, −3.02745224518697056068246818220, −2.75015643463081279282559512106, −2.68011911303121156072842646291, −2.61356352451505577272887869425, −2.50090587673871485689066299179, −2.14217644890542946699619683490, −2.03223047119631044791839955611, −1.90119766132551379067832184938, −1.79950512818282886491283119469, −1.65803099793457107463985576335, −1.10515141901177421920431499057, −1.02397296848201914162897612594, −1.01346280904488804929174107168, −0.75886969847794643049698866435, 0.75886969847794643049698866435, 1.01346280904488804929174107168, 1.02397296848201914162897612594, 1.10515141901177421920431499057, 1.65803099793457107463985576335, 1.79950512818282886491283119469, 1.90119766132551379067832184938, 2.03223047119631044791839955611, 2.14217644890542946699619683490, 2.50090587673871485689066299179, 2.61356352451505577272887869425, 2.68011911303121156072842646291, 2.75015643463081279282559512106, 3.02745224518697056068246818220, 3.03320621859166994473393779587, 3.09006779093623381268103510236, 3.59624359834717057980880729075, 3.59966196725121410229560625292, 3.71599298018287165749167569589, 3.87711647648630139783688324353, 3.90313934560955782230892078976, 3.98402774684297691284390846315, 4.16609295625747102122955072515, 4.18785379853488530109142632889, 4.52724933738373377326960140233

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

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