Properties

Label 8-882e4-1.1-c3e4-0-0
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $7.33396\times 10^{6}$
Root an. cond. $7.21385$
Motivic weight $3$
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 + 4·4-s + 16·8-s − 4·11-s − 64·16-s + 16·22-s − 60·23-s + 18·25-s + 848·29-s + 64·32-s − 492·37-s − 1.13e3·43-s − 16·44-s + 240·46-s − 72·50-s − 1.09e3·53-s − 3.39e3·58-s + 192·64-s − 1.30e3·67-s + 3.08e3·71-s + 1.96e3·74-s − 944·79-s + 4.54e3·86-s − 64·88-s − 240·92-s + 72·100-s + 4.38e3·106-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.707·8-s − 0.109·11-s − 16-s + 0.155·22-s − 0.543·23-s + 0.143·25-s + 5.42·29-s + 0.353·32-s − 2.18·37-s − 4.02·43-s − 0.0548·44-s + 0.769·46-s − 0.203·50-s − 2.84·53-s − 7.67·58-s + 3/8·64-s − 2.37·67-s + 5.14·71-s + 3.09·74-s − 1.34·79-s + 5.69·86-s − 0.0775·88-s − 0.271·92-s + 0.0719·100-s + 4.01·106-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(7.33396\times 10^{6}\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.002200279424\)
\(L(\frac12)\) \(\approx\) \(0.002200279424\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^3$ \( 1 - 18 T^{2} - 15301 T^{4} - 18 p^{6} T^{6} + p^{12} T^{8} \)
11$C_2^2$ \( ( 1 + 2 T - 1327 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 3466 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 7738 T^{2} + 35739075 T^{4} - 7738 p^{6} T^{6} + p^{12} T^{8} \)
19$C_2^3$ \( 1 + 9482 T^{2} + 42862443 T^{4} + 9482 p^{6} T^{6} + p^{12} T^{8} \)
23$C_2^2$ \( ( 1 + 30 T - 11267 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 212 T + p^{3} T^{2} )^{4} \)
31$C_2^3$ \( 1 - 14110 T^{2} - 688411581 T^{4} - 14110 p^{6} T^{6} + p^{12} T^{8} \)
37$C_2^2$ \( ( 1 + 246 T + 9863 T^{2} + 246 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 35530 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 284 T + p^{3} T^{2} )^{4} \)
47$C_2^3$ \( 1 - 203934 T^{2} + 30809861027 T^{4} - 203934 p^{6} T^{6} + p^{12} T^{8} \)
53$C_2^2$ \( ( 1 + 548 T + 151427 T^{2} + 548 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 38394 T^{2} - 40706434405 T^{4} + 38394 p^{6} T^{6} + p^{12} T^{8} \)
61$C_2^3$ \( 1 - 185770 T^{2} - 17009881461 T^{4} - 185770 p^{6} T^{6} + p^{12} T^{8} \)
67$C_2^2$ \( ( 1 + 652 T + 124341 T^{2} + 652 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 770 T + p^{3} T^{2} )^{4} \)
73$C_2^3$ \( 1 + 172238 T^{2} - 121668297645 T^{4} + 172238 p^{6} T^{6} + p^{12} T^{8} \)
79$C_2^2$ \( ( 1 + 472 T - 270255 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 1110166 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 897450 T^{2} + 308435211539 T^{4} - 897450 p^{6} T^{6} + p^{12} T^{8} \)
97$C_2^2$ \( ( 1 + 1732546 T^{2} + p^{6} T^{4} )^{2} \)
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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

−6.94418379319620600555524861359, −6.68272054655359403922975010316, −6.43600263913128821405581532515, −6.37231445600240265059460091363, −6.20117087815480231871551767231, −5.78973150129711513649653004042, −5.36984866013650187197803752032, −5.04628470369222934014733936784, −5.00715450074054127180921348179, −4.75241188660656020193570739131, −4.69727183591611758149629490894, −4.26379832998782045986602295769, −4.12089353917022782885285906226, −3.46283750014384151986841379199, −3.27303342996702108862046147078, −3.22551462921797305953963307235, −3.04238548924186821063007445591, −2.36661247360830379367372947633, −2.26785669854818127806268129549, −1.83841889631958519726045116508, −1.50967372765508650199969297650, −1.25859933387705786838903877597, −0.886721651065065868683968136443, −0.58051030137407099959542647424, −0.009707296996795882412279657498, 0.009707296996795882412279657498, 0.58051030137407099959542647424, 0.886721651065065868683968136443, 1.25859933387705786838903877597, 1.50967372765508650199969297650, 1.83841889631958519726045116508, 2.26785669854818127806268129549, 2.36661247360830379367372947633, 3.04238548924186821063007445591, 3.22551462921797305953963307235, 3.27303342996702108862046147078, 3.46283750014384151986841379199, 4.12089353917022782885285906226, 4.26379832998782045986602295769, 4.69727183591611758149629490894, 4.75241188660656020193570739131, 5.00715450074054127180921348179, 5.04628470369222934014733936784, 5.36984866013650187197803752032, 5.78973150129711513649653004042, 6.20117087815480231871551767231, 6.37231445600240265059460091363, 6.43600263913128821405581532515, 6.68272054655359403922975010316, 6.94418379319620600555524861359

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