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Properties

Label 8-3640e4-1.1-c0e4-0-10
Degree $8$
Conductor $1.756\times 10^{14}$
Sign $1$
Analytic cond. $10.8901$
Root an. cond. $1.34781$
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  − 2·4-s − 4·7-s + 3·16-s + 4·23-s + 8·28-s + 10·49-s − 4·64-s − 4·71-s − 2·81-s − 8·92-s − 12·112-s − 4·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 16·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·4-s − 4·7-s + 3·16-s + 4·23-s + 8·28-s + 10·49-s − 4·64-s − 4·71-s − 2·81-s − 8·92-s − 12·112-s − 4·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 16·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(10.8901\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3787287696\)
\(L(\frac12)\) \(\approx\) \(0.3787287696\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_1$ \( ( 1 + T )^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
good3$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + T^{4} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + T^{4} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{4} \)
71$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.35751034503684704572564124795, −6.03427720731406349479940032380, −5.78907278696809187039622068997, −5.66320602897418252644284118156, −5.31122734533838605751060168974, −5.25167748336550577929467742212, −5.14922990997045094871549941968, −4.97269642666772581237938583574, −4.46108783860133951336956182379, −4.33897283390852904243106070744, −4.18937128671485015765634143402, −3.98807145493428862521913242034, −3.77037656757199259756352342016, −3.54857816370319870024127680983, −3.28606660845439985351660578590, −3.16654056402469519808825711166, −2.87086701198958675238672594881, −2.78762071100878825862448186150, −2.60903410035937941984075114030, −2.54657914239734650665193809467, −1.54506892978470458269509045925, −1.37308496040701464666293346786, −1.25365407386316171078083363237, −0.53909406129678440978289457944, −0.46656766526877148816862920214, 0.46656766526877148816862920214, 0.53909406129678440978289457944, 1.25365407386316171078083363237, 1.37308496040701464666293346786, 1.54506892978470458269509045925, 2.54657914239734650665193809467, 2.60903410035937941984075114030, 2.78762071100878825862448186150, 2.87086701198958675238672594881, 3.16654056402469519808825711166, 3.28606660845439985351660578590, 3.54857816370319870024127680983, 3.77037656757199259756352342016, 3.98807145493428862521913242034, 4.18937128671485015765634143402, 4.33897283390852904243106070744, 4.46108783860133951336956182379, 4.97269642666772581237938583574, 5.14922990997045094871549941968, 5.25167748336550577929467742212, 5.31122734533838605751060168974, 5.66320602897418252644284118156, 5.78907278696809187039622068997, 6.03427720731406349479940032380, 6.35751034503684704572564124795

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