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Properties

Label 16-572e8-1.1-c1e8-0-0
Degree $16$
Conductor $1.146\times 10^{22}$
Sign $1$
Analytic cond. $189402.$
Root an. cond. $2.13715$
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  − 2·3-s − 9-s − 8·13-s + 16·17-s + 10·23-s − 6·25-s − 4·27-s − 4·29-s + 16·39-s − 20·43-s + 29·49-s − 32·51-s + 38·53-s − 36·61-s − 20·69-s + 12·75-s + 40·79-s + 11·81-s + 8·87-s − 48·101-s + 18·103-s − 28·107-s + 58·113-s + 8·117-s − 4·121-s + 127-s + 40·129-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/3·9-s − 2.21·13-s + 3.88·17-s + 2.08·23-s − 6/5·25-s − 0.769·27-s − 0.742·29-s + 2.56·39-s − 3.04·43-s + 29/7·49-s − 4.48·51-s + 5.21·53-s − 4.60·61-s − 2.40·69-s + 1.38·75-s + 4.50·79-s + 11/9·81-s + 0.857·87-s − 4.77·101-s + 1.77·103-s − 2.70·107-s + 5.45·113-s + 0.739·117-s − 0.363·121-s + 0.0887·127-s + 3.52·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 11^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(189402.\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 11^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.295013463\)
\(L(\frac12)\) \(\approx\) \(1.295013463\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( ( 1 + T^{2} )^{4} \)
13 \( 1 + 8 T + 18 T^{2} + 8 T^{3} + 2 T^{4} + 8 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
good3 \( ( 1 + T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
5 \( 1 + 6 T^{2} + 49 T^{4} + 306 T^{6} + 348 p T^{8} + 306 p^{2} T^{10} + 49 p^{4} T^{12} + 6 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 - 29 T^{2} + 407 T^{4} - 3831 T^{6} + 28856 T^{8} - 3831 p^{2} T^{10} + 407 p^{4} T^{12} - 29 p^{6} T^{14} + p^{8} T^{16} \)
17 \( ( 1 - 8 T + 54 T^{2} - 192 T^{3} + 906 T^{4} - 192 p T^{5} + 54 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
19 \( 1 - 45 T^{2} + 1551 T^{4} - 38759 T^{6} + 815964 T^{8} - 38759 p^{2} T^{10} + 1551 p^{4} T^{12} - 45 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 - 5 T + 64 T^{2} - 222 T^{3} + 1775 T^{4} - 222 p T^{5} + 64 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 2 T + 46 T^{2} - 166 T^{3} + 642 T^{4} - 166 p T^{5} + 46 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 - 58 T^{2} + 1945 T^{4} - 82822 T^{6} + 3288412 T^{8} - 82822 p^{2} T^{10} + 1945 p^{4} T^{12} - 58 p^{6} T^{14} + p^{8} T^{16} \)
37 \( 1 - 174 T^{2} + 16065 T^{4} - 26390 p T^{6} + 42457908 T^{8} - 26390 p^{3} T^{10} + 16065 p^{4} T^{12} - 174 p^{6} T^{14} + p^{8} T^{16} \)
41 \( 1 - 53 T^{2} + 115 p T^{4} - 193263 T^{6} + 10976756 T^{8} - 193263 p^{2} T^{10} + 115 p^{5} T^{12} - 53 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 + 10 T + 138 T^{2} + 726 T^{3} + 6898 T^{4} + 726 p T^{5} + 138 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 244 T^{2} + 29108 T^{4} - 2252748 T^{6} + 124096406 T^{8} - 2252748 p^{2} T^{10} + 29108 p^{4} T^{12} - 244 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 - 19 T + 271 T^{2} - 2733 T^{3} + 23324 T^{4} - 2733 p T^{5} + 271 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 206 T^{2} + 19241 T^{4} - 1196838 T^{6} + 67809284 T^{8} - 1196838 p^{2} T^{10} + 19241 p^{4} T^{12} - 206 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 + 18 T + 294 T^{2} + 2746 T^{3} + 426 p T^{4} + 2746 p T^{5} + 294 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 314 T^{2} + 49337 T^{4} - 5171766 T^{6} + 398651756 T^{8} - 5171766 p^{2} T^{10} + 49337 p^{4} T^{12} - 314 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 - 250 T^{2} + 27065 T^{4} - 1845414 T^{6} + 117329020 T^{8} - 1845414 p^{2} T^{10} + 27065 p^{4} T^{12} - 250 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 265 T^{2} + 44059 T^{4} - 4880443 T^{6} + 414776080 T^{8} - 4880443 p^{2} T^{10} + 44059 p^{4} T^{12} - 265 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 20 T + 314 T^{2} - 3732 T^{3} + 39586 T^{4} - 3732 p T^{5} + 314 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 457 T^{2} + 101207 T^{4} - 14199027 T^{6} + 1391424476 T^{8} - 14199027 p^{2} T^{10} + 101207 p^{4} T^{12} - 457 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 - 266 T^{2} + 29505 T^{4} - 1036110 T^{6} - 9499940 T^{8} - 1036110 p^{2} T^{10} + 29505 p^{4} T^{12} - 266 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 - 474 T^{2} + 115649 T^{4} - 18494574 T^{6} + 2107503900 T^{8} - 18494574 p^{2} T^{10} + 115649 p^{4} T^{12} - 474 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.73857705102577970216810986047, −4.60263593682069833882933190920, −4.50503015871190510782457207083, −4.40346387346905002073277618754, −4.00999326514715338167956485176, −3.90788565597923771435762348815, −3.79669096866697468628436107133, −3.71765883125769909605392259340, −3.71099589811184224271222357323, −3.43024853924169661377162810687, −3.20172384058522593607306585582, −3.13786318747423411064850628201, −2.93051507855104843744430943545, −2.74679088089360158355478735013, −2.53876264786716688172336186594, −2.49037522587315393323109036326, −2.30598217705380690069498805466, −2.11485173961056099699713367907, −1.84221215028483669303900888342, −1.62390971772423844438212996444, −1.21841841103577539839715012567, −1.19924437337209641535381475414, −1.03017185286626930310215862552, −0.45659696260437373711446877668, −0.32031759698496543665892156220, 0.32031759698496543665892156220, 0.45659696260437373711446877668, 1.03017185286626930310215862552, 1.19924437337209641535381475414, 1.21841841103577539839715012567, 1.62390971772423844438212996444, 1.84221215028483669303900888342, 2.11485173961056099699713367907, 2.30598217705380690069498805466, 2.49037522587315393323109036326, 2.53876264786716688172336186594, 2.74679088089360158355478735013, 2.93051507855104843744430943545, 3.13786318747423411064850628201, 3.20172384058522593607306585582, 3.43024853924169661377162810687, 3.71099589811184224271222357323, 3.71765883125769909605392259340, 3.79669096866697468628436107133, 3.90788565597923771435762348815, 4.00999326514715338167956485176, 4.40346387346905002073277618754, 4.50503015871190510782457207083, 4.60263593682069833882933190920, 4.73857705102577970216810986047

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

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