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Properties

Label 16-4032e8-1.1-c1e8-0-6
Degree $16$
Conductor $6.985\times 10^{28}$
Sign $1$
Analytic cond. $1.15446\times 10^{12}$
Root an. cond. $5.67412$
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  + 4·7-s − 8·19-s + 12·25-s − 16·31-s − 8·37-s − 16·47-s + 8·49-s + 16·53-s + 32·59-s + 16·83-s − 48·103-s + 20·121-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 48·175-s + 179-s + 181-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.83·19-s + 12/5·25-s − 2.87·31-s − 1.31·37-s − 2.33·47-s + 8/7·49-s + 2.19·53-s + 4.16·59-s + 1.75·83-s − 4.72·103-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 + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + 0.0760·173-s + 3.62·175-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.15446\times 10^{12}\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.972249730\)
\(L(\frac12)\) \(\approx\) \(3.972249730\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−3.52401175744270656387813305564, −3.37415717735353594009020786304, −3.28128901830693339618305091416, −3.10629923907987206345258962080, −2.86053297741293849446757638245, −2.85685289846444022902768906882, −2.83954013573704678088389114091, −2.80631426209119781256646187835, −2.64864288892857368652685498841, −2.19584967488733657070802555864, −2.17547177301721359048092209849, −2.01609477966049875933764726286, −2.01313024033400201475429234170, −1.97233472988403162082796243801, −1.86883695930919853426939847607, −1.84652889593333714789830182542, −1.46066185824358006984345042888, −1.40452934593673066416288806614, −1.36200658278277885533344074903, −0.946390183662666333045840668658, −0.77606582142321662862091261809, −0.70363953043353346940643913160, −0.69872793542977972406389779136, −0.41518580245653130349598450888, −0.11551815526513963941706193184, 0.11551815526513963941706193184, 0.41518580245653130349598450888, 0.69872793542977972406389779136, 0.70363953043353346940643913160, 0.77606582142321662862091261809, 0.946390183662666333045840668658, 1.36200658278277885533344074903, 1.40452934593673066416288806614, 1.46066185824358006984345042888, 1.84652889593333714789830182542, 1.86883695930919853426939847607, 1.97233472988403162082796243801, 2.01313024033400201475429234170, 2.01609477966049875933764726286, 2.17547177301721359048092209849, 2.19584967488733657070802555864, 2.64864288892857368652685498841, 2.80631426209119781256646187835, 2.83954013573704678088389114091, 2.85685289846444022902768906882, 2.86053297741293849446757638245, 3.10629923907987206345258962080, 3.28128901830693339618305091416, 3.37415717735353594009020786304, 3.52401175744270656387813305564

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

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