Properties

Label 16-99e8-1.1-c1e8-0-2
Degree $16$
Conductor $9.227\times 10^{15}$
Sign $1$
Analytic cond. $0.152510$
Root an. cond. $0.889111$
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  + 3·4-s − 6·7-s − 6·13-s + 9·16-s + 12·19-s + 9·25-s − 18·28-s + 12·31-s − 18·37-s − 32·43-s + 3·49-s − 18·52-s + 42·61-s + 25·64-s + 4·67-s + 24·73-s + 36·76-s + 30·79-s + 36·91-s − 54·97-s + 27·100-s − 36·103-s − 56·109-s − 54·112-s + 19·121-s + 36·124-s + 127-s + ⋯
L(s)  = 1  + 3/2·4-s − 2.26·7-s − 1.66·13-s + 9/4·16-s + 2.75·19-s + 9/5·25-s − 3.40·28-s + 2.15·31-s − 2.95·37-s − 4.87·43-s + 3/7·49-s − 2.49·52-s + 5.37·61-s + 25/8·64-s + 0.488·67-s + 2.80·73-s + 4.12·76-s + 3.37·79-s + 3.77·91-s − 5.48·97-s + 2.69·100-s − 3.54·103-s − 5.36·109-s − 5.10·112-s + 1.72·121-s + 3.23·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(0.152510\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9704985766\)
\(L(\frac12)\) \(\approx\) \(0.9704985766\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 - 19 T^{2} + 21 p T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \)
good2 \( 1 - 3 T^{2} + p T^{6} + 9 T^{8} + p^{3} T^{10} - 3 p^{6} T^{14} + p^{8} T^{16} \)
5 \( 1 - 9 T^{2} + 21 T^{4} + 161 T^{6} - 1644 T^{8} + 161 p^{2} T^{10} + 21 p^{4} T^{12} - 9 p^{6} T^{14} + p^{8} T^{16} \)
7 \( ( 1 + 3 T + 12 T^{2} + 5 p T^{3} + 141 T^{4} + 5 p^{2} T^{5} + 12 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 3 T + 6 T^{2} + 59 T^{3} + 339 T^{4} + 59 p T^{5} + 6 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 - 279 T^{4} - 1420 T^{6} + 82101 T^{8} - 1420 p^{2} T^{10} - 279 p^{4} T^{12} + p^{8} T^{16} \)
19 \( ( 1 - 6 T - 3 T^{2} + 122 T^{3} - 525 T^{4} + 122 p T^{5} - 3 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 20 T^{2} + 753 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( 1 + 18 T^{2} + 1683 T^{4} + 10856 T^{6} + 1510005 T^{8} + 10856 p^{2} T^{10} + 1683 p^{4} T^{12} + 18 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 - 6 T - 15 T^{2} + 206 T^{3} - 561 T^{4} + 206 p T^{5} - 15 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 9 T - 6 T^{2} - 307 T^{3} - 1581 T^{4} - 307 p T^{5} - 6 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 87 T^{2} + 2538 T^{4} + 70841 T^{6} - 7729545 T^{8} + 70841 p^{2} T^{10} + 2538 p^{4} T^{12} - 87 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
47 \( 1 - 75 T^{2} + 5541 T^{4} - 375925 T^{6} + 17183556 T^{8} - 375925 p^{2} T^{10} + 5541 p^{4} T^{12} - 75 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 24 T^{2} + 5817 T^{4} - 20908 T^{6} + 15682005 T^{8} - 20908 p^{2} T^{10} + 5817 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 - 72 T^{2} - 327 T^{4} + 271796 T^{6} - 16865475 T^{8} + 271796 p^{2} T^{10} - 327 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 - 21 T + 135 T^{2} - 259 T^{3} + 144 T^{4} - 259 p T^{5} + 135 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - T + 123 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \)
71 \( 1 - 117 T^{2} + 273 T^{4} + 566981 T^{6} - 45119220 T^{8} + 566981 p^{2} T^{10} + 273 p^{4} T^{12} - 117 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 - 12 T - 9 T^{2} + 164 T^{3} + 3249 T^{4} + 164 p T^{5} - 9 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 15 T + 6 T^{2} + 815 T^{3} - 5979 T^{4} + 815 p T^{5} + 6 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 177 T^{2} + 9810 T^{4} + 724223 T^{6} - 144542001 T^{8} + 724223 p^{2} T^{10} + 9810 p^{4} T^{12} - 177 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 4 T^{2} + 5721 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 27 T + 357 T^{2} + 4325 T^{3} + 49476 T^{4} + 4325 p T^{5} + 357 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−6.76185038977254281236464950227, −6.43910264623084145944378702522, −6.33610167744056927485100486823, −6.27730222672653421828775026783, −6.07301908876095571532798415771, −5.62712327188243204286702406973, −5.24932177805359400080755508986, −5.23476786268654704783853889482, −5.14267665302715992209516976583, −5.14152842929907872875689142425, −5.11402252571010367400531264764, −4.99066353992131693364620669113, −4.23769931370944255059205384103, −3.96728289859875483370294704413, −3.80777498854078035110093893622, −3.76683826341343913067536598885, −3.41797442981747160207817098763, −3.13160208856049619505524521450, −3.05229173116034381947425397531, −2.96592272574710577737959721356, −2.57871003985398342127592857640, −2.48707916072600681071858994349, −1.97544366425961921616310001188, −1.54587090049141863590163174685, −1.07313258794831466183241338671, 1.07313258794831466183241338671, 1.54587090049141863590163174685, 1.97544366425961921616310001188, 2.48707916072600681071858994349, 2.57871003985398342127592857640, 2.96592272574710577737959721356, 3.05229173116034381947425397531, 3.13160208856049619505524521450, 3.41797442981747160207817098763, 3.76683826341343913067536598885, 3.80777498854078035110093893622, 3.96728289859875483370294704413, 4.23769931370944255059205384103, 4.99066353992131693364620669113, 5.11402252571010367400531264764, 5.14152842929907872875689142425, 5.14267665302715992209516976583, 5.23476786268654704783853889482, 5.24932177805359400080755508986, 5.62712327188243204286702406973, 6.07301908876095571532798415771, 6.27730222672653421828775026783, 6.33610167744056927485100486823, 6.43910264623084145944378702522, 6.76185038977254281236464950227

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

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