Properties

Label 16-1560e8-1.1-c0e8-0-0
Degree $16$
Conductor $3.507\times 10^{25}$
Sign $1$
Analytic cond. $0.134975$
Root an. cond. $0.882349$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·43-s − 2·81-s − 8·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  − 8·43-s − 2·81-s − 8·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + 251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(0.134975\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004507173048\)
\(L(\frac12)\) \(\approx\) \(0.004507173048\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{8} \)
3 \( ( 1 + T^{4} )^{2} \)
5 \( 1 + T^{8} \)
13 \( ( 1 + T^{4} )^{2} \)
good7 \( ( 1 + T^{4} )^{4} \)
11 \( ( 1 + T^{8} )^{2} \)
17 \( ( 1 + T^{4} )^{4} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 + T^{4} )^{4} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 + T^{2} )^{8} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 + T^{8} )^{2} \)
43 \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \)
47 \( ( 1 + T^{8} )^{2} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 + T^{8} )^{2} \)
61 \( ( 1 + T^{4} )^{4} \)
67 \( ( 1 + T^{4} )^{4} \)
71 \( ( 1 + T^{8} )^{2} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 + T^{4} )^{4} \)
83 \( ( 1 + T^{8} )^{2} \)
89 \( ( 1 + T^{8} )^{2} \)
97 \( ( 1 + T^{4} )^{4} \)
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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

−4.23684759866024398917128283392, −4.21844466512772928948418280886, −3.90365071498691143984784925395, −3.85871012120193172970736729992, −3.67899003066210673536977811417, −3.58320074206911322882441922879, −3.47981297561342951219708081132, −3.44225802900443672317375322903, −3.17222457461606376476687522728, −3.01605514531165697726192175769, −2.99144432707594558910151211399, −2.88916611630815101354547698073, −2.66010211980967207248313129505, −2.62207775017757905383560083936, −2.36732724401881012088599707655, −2.32081276718488643788105210728, −2.06177190652976047698193965960, −1.82634627406202565527851043338, −1.67722487981037798104266740889, −1.60209411274406067817478018437, −1.53247708731844805900247322929, −1.28207386368279233686040014983, −1.15159301427133153853303145341, −0.959729820655317169865916482465, −0.02752846837848939101940541531, 0.02752846837848939101940541531, 0.959729820655317169865916482465, 1.15159301427133153853303145341, 1.28207386368279233686040014983, 1.53247708731844805900247322929, 1.60209411274406067817478018437, 1.67722487981037798104266740889, 1.82634627406202565527851043338, 2.06177190652976047698193965960, 2.32081276718488643788105210728, 2.36732724401881012088599707655, 2.62207775017757905383560083936, 2.66010211980967207248313129505, 2.88916611630815101354547698073, 2.99144432707594558910151211399, 3.01605514531165697726192175769, 3.17222457461606376476687522728, 3.44225802900443672317375322903, 3.47981297561342951219708081132, 3.58320074206911322882441922879, 3.67899003066210673536977811417, 3.85871012120193172970736729992, 3.90365071498691143984784925395, 4.21844466512772928948418280886, 4.23684759866024398917128283392

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

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