Properties

Label 8-96e8-1.1-c1e4-0-15
Degree $8$
Conductor $7.214\times 10^{15}$
Sign $1$
Analytic cond. $2.93277\times 10^{7}$
Root an. cond. $8.57846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 4·7-s − 8·13-s + 12·29-s − 12·31-s − 16·35-s − 16·37-s − 4·49-s + 20·53-s − 16·61-s − 32·65-s − 16·67-s − 8·71-s − 8·73-s − 12·79-s + 8·89-s + 32·91-s + 20·101-s − 4·103-s − 24·109-s + 8·113-s − 20·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.51·7-s − 2.21·13-s + 2.22·29-s − 2.15·31-s − 2.70·35-s − 2.63·37-s − 4/7·49-s + 2.74·53-s − 2.04·61-s − 3.96·65-s − 1.95·67-s − 0.949·71-s − 0.936·73-s − 1.35·79-s + 0.847·89-s + 3.35·91-s + 1.99·101-s − 0.394·103-s − 2.29·109-s + 0.752·113-s − 1.81·121-s − 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(2.93277\times 10^{7}\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{40} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 16 T^{2} - 44 T^{3} + 118 T^{4} - 44 p T^{5} + 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 20 T^{2} + 60 T^{3} + 186 T^{4} + 60 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 20 T^{2} - 32 T^{3} + 230 T^{4} - 32 p T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 56 T^{2} + 264 T^{3} + 1122 T^{4} + 264 p T^{5} + 56 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 36 T^{2} + 64 T^{3} + 662 T^{4} + 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 44 T^{2} + 64 T^{3} + 966 T^{4} + 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 144 T^{2} - 964 T^{3} + 6422 T^{4} - 964 p T^{5} + 144 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 164 T^{2} + 1140 T^{3} + 8218 T^{4} + 1140 p T^{5} + 164 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 200 T^{2} + 1552 T^{3} + 11010 T^{4} + 1552 p T^{5} + 200 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 100 T^{2} - 192 T^{3} + 4726 T^{4} - 192 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 76 T^{2} + 256 T^{3} + 2726 T^{4} + 256 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 336 T^{2} - 3452 T^{3} + 30134 T^{4} - 3452 p T^{5} + 336 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 296 T^{2} + 2704 T^{3} + 27618 T^{4} + 2704 p T^{5} + 296 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 4 p T^{2} + 2960 T^{3} + 27190 T^{4} + 2960 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 252 T^{2} + 1512 T^{3} + 25766 T^{4} + 1512 p T^{5} + 252 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 196 T^{2} + 1816 T^{3} + 18022 T^{4} + 1816 p T^{5} + 196 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 148 T^{2} - 44 T^{3} + 794 T^{4} - 44 p T^{5} + 148 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 116 T^{2} - 160 T^{3} + 5510 T^{4} - 160 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 156 T^{2} - 504 T^{3} + 10022 T^{4} - 504 p T^{5} + 156 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 768 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \)
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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

−5.91471120949841120213277089564, −5.52605305101481367746918044508, −5.44300765590518981219329346032, −5.27451504138362658809828965842, −5.18284418580208290840199427414, −4.79545941862084288753831843483, −4.70200196402395204152806714631, −4.60533790497788386929244934669, −4.48893118421042740989405205754, −4.06918700421161490530044898343, −3.84157716254594293588124075934, −3.67232729341170555968236840071, −3.59803977904703061553196229641, −3.21597994206012395271103715243, −3.10548236404279331550853865048, −2.95636127247294772593005024131, −2.70880450164904797332739984038, −2.46261248375126476298714136915, −2.15622746189881451363511879624, −2.15066212180681415695123890908, −2.14228024086616351413874294596, −1.51217645762594671794788018692, −1.45088922218785100626924032811, −1.18429459657222132940059979075, −1.01786846379568371680924840315, 0, 0, 0, 0, 1.01786846379568371680924840315, 1.18429459657222132940059979075, 1.45088922218785100626924032811, 1.51217645762594671794788018692, 2.14228024086616351413874294596, 2.15066212180681415695123890908, 2.15622746189881451363511879624, 2.46261248375126476298714136915, 2.70880450164904797332739984038, 2.95636127247294772593005024131, 3.10548236404279331550853865048, 3.21597994206012395271103715243, 3.59803977904703061553196229641, 3.67232729341170555968236840071, 3.84157716254594293588124075934, 4.06918700421161490530044898343, 4.48893118421042740989405205754, 4.60533790497788386929244934669, 4.70200196402395204152806714631, 4.79545941862084288753831843483, 5.18284418580208290840199427414, 5.27451504138362658809828965842, 5.44300765590518981219329346032, 5.52605305101481367746918044508, 5.91471120949841120213277089564

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