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Properties

Label 8-1560e4-1.1-c0e4-0-1
Degree $8$
Conductor $5.922\times 10^{12}$
Sign $1$
Analytic cond. $0.367389$
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  − 2·9-s − 16-s + 4·17-s + 4·43-s + 3·81-s + 4·107-s − 4·113-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s − 8·153-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2·9-s − 16-s + 4·17-s + 4·43-s + 3·81-s + 4·107-s − 4·113-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s − 8·153-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(0.367389\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100587597\)
\(L(\frac12)\) \(\approx\) \(1.100587597\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
3$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2^2$ \( 1 + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
good7$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_2^2$ \( ( 1 + T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−6.89651324810535105516244453191, −6.67614029781394825009949343312, −6.45876255872933951486078947224, −6.23667599481170632077665688881, −5.93931269872367222149634940592, −5.81336747421711330184063362126, −5.67056732946157086730293372168, −5.52238146006170480284387723841, −5.18128536952878359356792499441, −5.06808002884782947531880488265, −5.01866581593170072494926261589, −4.32909344991039553932524331509, −4.26530475305542719394515418278, −4.06919068177439757629953234196, −3.76595838011834128611490361584, −3.36033630425602045094744731035, −3.29981189407777004092295128691, −3.03073712906181149412029151633, −2.78567125768940623919766862562, −2.53586982846714535759586505236, −2.25147688489746114484075017064, −1.98583808909537944933566435786, −1.34772033974007551188274029329, −1.07937541647833541594662425512, −0.74045425211659851367426791725, 0.74045425211659851367426791725, 1.07937541647833541594662425512, 1.34772033974007551188274029329, 1.98583808909537944933566435786, 2.25147688489746114484075017064, 2.53586982846714535759586505236, 2.78567125768940623919766862562, 3.03073712906181149412029151633, 3.29981189407777004092295128691, 3.36033630425602045094744731035, 3.76595838011834128611490361584, 4.06919068177439757629953234196, 4.26530475305542719394515418278, 4.32909344991039553932524331509, 5.01866581593170072494926261589, 5.06808002884782947531880488265, 5.18128536952878359356792499441, 5.52238146006170480284387723841, 5.67056732946157086730293372168, 5.81336747421711330184063362126, 5.93931269872367222149634940592, 6.23667599481170632077665688881, 6.45876255872933951486078947224, 6.67614029781394825009949343312, 6.89651324810535105516244453191

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