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Properties

Label 16-2268e8-1.1-c0e8-0-0
Degree $16$
Conductor $7.001\times 10^{26}$
Sign $1$
Analytic cond. $2.69402$
Root an. cond. $1.06389$
Motivic weight $0$
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-s + 4·25-s + 2·49-s − 12·67-s − 12·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 16-s + 4·25-s + 2·49-s − 12·67-s − 12·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{32} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.69402\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{32} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06188646437\)
\(L(\frac12)\) \(\approx\) \(0.06188646437\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{4} + T^{8} \)
3 \( 1 \)
7 \( ( 1 - T^{2} + T^{4} )^{2} \)
good5 \( ( 1 - T^{2} + T^{4} )^{4} \)
11 \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
13 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
17 \( ( 1 + T^{2} )^{8} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 - T^{4} + T^{8} )^{2} \)
29 \( ( 1 - T^{4} + T^{8} )^{2} \)
31 \( ( 1 - T^{2} + T^{4} )^{4} \)
37 \( ( 1 - T^{2} + T^{4} )^{4} \)
41 \( ( 1 - T^{2} + T^{4} )^{4} \)
43 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
47 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
53 \( ( 1 - T^{4} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
61 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
67 \( ( 1 + T )^{8}( 1 + T + T^{2} )^{4} \)
71 \( ( 1 - T^{4} + T^{8} )^{2} \)
73 \( ( 1 - T )^{8}( 1 + T )^{8} \)
79 \( ( 1 + T )^{8}( 1 + T + T^{2} )^{4} \)
83 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
89 \( ( 1 + T^{2} )^{8} \)
97 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
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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

−4.10224573903107779714782927856, −4.03588064868000120881986775735, −3.78205977378399346414529348844, −3.45972398928758771373532980601, −3.44763913704678022493756135255, −3.38703637044892145761383098612, −3.10619868684909420804412692441, −3.03120966339614864431093644634, −3.01440143136858838978983986230, −2.96126750358873915786053515832, −2.88613039795436815491106179330, −2.67670217548251875894349887580, −2.61033707195885073236433122931, −2.58807203377252978256114020624, −2.28351530978771990606124622162, −2.07380927584292872184408146141, −1.88603586457075514061258982568, −1.63264614810610308897662031210, −1.55844927099671325418970651658, −1.32470931576504892362420045390, −1.32245215164252951105350974779, −1.30464371774011452369715669469, −1.07282001985636962221338768360, −0.929682143265937135147022561357, −0.07181240178406111150180165859, 0.07181240178406111150180165859, 0.929682143265937135147022561357, 1.07282001985636962221338768360, 1.30464371774011452369715669469, 1.32245215164252951105350974779, 1.32470931576504892362420045390, 1.55844927099671325418970651658, 1.63264614810610308897662031210, 1.88603586457075514061258982568, 2.07380927584292872184408146141, 2.28351530978771990606124622162, 2.58807203377252978256114020624, 2.61033707195885073236433122931, 2.67670217548251875894349887580, 2.88613039795436815491106179330, 2.96126750358873915786053515832, 3.01440143136858838978983986230, 3.03120966339614864431093644634, 3.10619868684909420804412692441, 3.38703637044892145761383098612, 3.44763913704678022493756135255, 3.45972398928758771373532980601, 3.78205977378399346414529348844, 4.03588064868000120881986775735, 4.10224573903107779714782927856

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

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