Properties

Label 8-1575e4-1.1-c0e4-0-4
Degree $8$
Conductor $6.154\times 10^{12}$
Sign $1$
Analytic cond. $0.381725$
Root an. cond. $0.886581$
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  + 4·2-s + 8·4-s + 12·8-s + 18·16-s − 4·23-s + 28·32-s − 16·46-s − 4·53-s + 40·64-s − 32·92-s − 16·106-s + 4·107-s + 4·113-s + 127-s + 52·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 48·184-s + ⋯
L(s)  = 1  + 4·2-s + 8·4-s + 12·8-s + 18·16-s − 4·23-s + 28·32-s − 16·46-s − 4·53-s + 40·64-s − 32·92-s − 16·106-s + 4·107-s + 4·113-s + 127-s + 52·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 48·184-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.381725\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(12.32249843\)
\(L(\frac12)\) \(\approx\) \(12.32249843\)
\(L(1)\) 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)$
bad3 \( 1 \)
5 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good2$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
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

−6.56215842585024997409073439737, −6.46141937531656666784095877689, −6.24250962183797518004255215219, −6.09312313545987489851557351538, −5.87062846748685346960231179231, −5.86943550999571355956861441954, −5.70434431063958509331185228818, −5.16386153263934164796902579443, −4.99473716344798591472882486957, −4.92000633166331703893434148320, −4.85393285231837693529102005690, −4.40736471351205059243657853202, −4.24405567623656101125730488152, −4.15706526693404224822204476378, −3.97280998433675398358814642686, −3.64624729068041528649182521423, −3.28527492964716103587803528489, −3.26212981803990160417194471414, −3.25025122499728783537351182356, −2.66671426068583057859247343096, −2.24318965791129393154419731591, −1.99853543848525930454647359952, −1.92967122841025995921833626643, −1.42859601942606664496582095775, −1.09574911540786564503374128921, 1.09574911540786564503374128921, 1.42859601942606664496582095775, 1.92967122841025995921833626643, 1.99853543848525930454647359952, 2.24318965791129393154419731591, 2.66671426068583057859247343096, 3.25025122499728783537351182356, 3.26212981803990160417194471414, 3.28527492964716103587803528489, 3.64624729068041528649182521423, 3.97280998433675398358814642686, 4.15706526693404224822204476378, 4.24405567623656101125730488152, 4.40736471351205059243657853202, 4.85393285231837693529102005690, 4.92000633166331703893434148320, 4.99473716344798591472882486957, 5.16386153263934164796902579443, 5.70434431063958509331185228818, 5.86943550999571355956861441954, 5.87062846748685346960231179231, 6.09312313545987489851557351538, 6.24250962183797518004255215219, 6.46141937531656666784095877689, 6.56215842585024997409073439737

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