Properties

Label 8-4608e4-1.1-c1e4-0-34
Degree $8$
Conductor $4.509\times 10^{14}$
Sign $1$
Analytic cond. $1.83298\times 10^{6}$
Root an. cond. $6.06589$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s + 32·25-s − 8·29-s − 40·53-s − 8·97-s − 56·101-s − 20·121-s − 104·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 3.57·5-s + 32/5·25-s − 1.48·29-s − 5.49·53-s − 0.812·97-s − 5.57·101-s − 1.81·121-s − 9.30·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.83298\times 10^{6}\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4608} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{36} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
7$C_2^3$ \( 1 + 2 T^{4} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^3$ \( 1 - 478 T^{4} + p^{4} T^{8} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^3$ \( 1 - 9118 T^{4} + p^{4} T^{8} \)
83$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
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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

−6.43592784499019734119024227716, −6.05235057756333677023197321143, −5.86814817996398070544519789692, −5.66102023336549444773855201943, −5.41815457260965965954292012282, −5.08704173172540960328299800431, −5.06093811037283114694691485451, −4.88824775076416887208885082597, −4.59896907418805351579380447249, −4.28365057850080691351707872323, −4.27483619460294545903838580498, −4.17697399318567713829293741196, −3.91347802080894522581071908547, −3.57082452740568230864889547947, −3.50167902742170212072248604580, −3.30984004833142965690932655982, −3.23957931095798611527466420214, −2.97577267528829505546894701006, −2.47619177008046830267795510603, −2.37320861918770211587373865325, −2.32338197298546205039994443979, −1.52643768638780490220007532166, −1.43598925917352209943207387673, −1.25332360563874722025596108827, −0.987705560611143327129337532796, 0, 0, 0, 0, 0.987705560611143327129337532796, 1.25332360563874722025596108827, 1.43598925917352209943207387673, 1.52643768638780490220007532166, 2.32338197298546205039994443979, 2.37320861918770211587373865325, 2.47619177008046830267795510603, 2.97577267528829505546894701006, 3.23957931095798611527466420214, 3.30984004833142965690932655982, 3.50167902742170212072248604580, 3.57082452740568230864889547947, 3.91347802080894522581071908547, 4.17697399318567713829293741196, 4.27483619460294545903838580498, 4.28365057850080691351707872323, 4.59896907418805351579380447249, 4.88824775076416887208885082597, 5.06093811037283114694691485451, 5.08704173172540960328299800431, 5.41815457260965965954292012282, 5.66102023336549444773855201943, 5.86814817996398070544519789692, 6.05235057756333677023197321143, 6.43592784499019734119024227716

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