Properties

Label 8-9800e4-1.1-c1e4-0-7
Degree $8$
Conductor $9.224\times 10^{15}$
Sign $1$
Analytic cond. $3.74983\times 10^{7}$
Root an. cond. $8.84609$
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  + 3·3-s + 9-s − 4·13-s − 7·17-s − 10·19-s − 5·27-s + 2·29-s − 14·31-s − 2·37-s − 12·39-s − 4·41-s + 15·43-s − 15·47-s − 21·51-s − 10·53-s − 30·57-s − 59-s − 25·61-s − 4·67-s − 20·71-s − 2·73-s + 2·79-s − 81-s + 26·83-s + 6·87-s − 19·89-s − 42·93-s + ⋯
L(s)  = 1  + 1.73·3-s + 1/3·9-s − 1.10·13-s − 1.69·17-s − 2.29·19-s − 0.962·27-s + 0.371·29-s − 2.51·31-s − 0.328·37-s − 1.92·39-s − 0.624·41-s + 2.28·43-s − 2.18·47-s − 2.94·51-s − 1.37·53-s − 3.97·57-s − 0.130·59-s − 3.20·61-s − 0.488·67-s − 2.37·71-s − 0.234·73-s + 0.225·79-s − 1/9·81-s + 2.85·83-s + 0.643·87-s − 2.01·89-s − 4.35·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3.74983\times 10^{7}\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3$C_2 \wr S_4$ \( 1 - p T + 8 T^{2} - 16 T^{3} + 26 T^{4} - 16 p T^{5} + 8 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 7 T^{2} - 49 T^{3} + 46 T^{4} - 49 p T^{5} + 7 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 4 T + 29 T^{2} + 51 T^{3} + 332 T^{4} + 51 p T^{5} + 29 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 7 T + 75 T^{2} + 339 T^{3} + 1977 T^{4} + 339 p T^{5} + 75 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 10 T + 63 T^{2} + 211 T^{3} + 848 T^{4} + 211 p T^{5} + 63 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 13 T^{2} - 91 T^{3} + 493 T^{4} - 91 p T^{5} + 13 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 2 T + 67 T^{2} + 45 T^{3} + 1982 T^{4} + 45 p T^{5} + 67 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 14 T + 173 T^{2} + 1299 T^{3} + 8779 T^{4} + 1299 p T^{5} + 173 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 2 T + 105 T^{2} + 95 T^{3} + 4940 T^{4} + 95 p T^{5} + 105 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 4 T + 37 T^{2} + 367 T^{3} + 2991 T^{4} + 367 p T^{5} + 37 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 15 T + 176 T^{2} - 1314 T^{3} + 9482 T^{4} - 1314 p T^{5} + 176 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 15 T + 189 T^{2} + 1299 T^{3} + 10333 T^{4} + 1299 p T^{5} + 189 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 10 T + 241 T^{2} + 1603 T^{3} + 19896 T^{4} + 1603 p T^{5} + 241 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + T + 170 T^{2} + 100 T^{3} + 13168 T^{4} + 100 p T^{5} + 170 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 25 T + 408 T^{2} + 4518 T^{3} + 40204 T^{4} + 4518 p T^{5} + 408 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 4 T + 96 T^{2} - 212 T^{3} + 2414 T^{4} - 212 p T^{5} + 96 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 20 T + 318 T^{2} + 3528 T^{3} + 34039 T^{4} + 3528 p T^{5} + 318 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 2 T + 108 T^{2} + 54 T^{3} + 6262 T^{4} + 54 p T^{5} + 108 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 2 T + 241 T^{2} - 209 T^{3} + 25447 T^{4} - 209 p T^{5} + 241 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 26 T + 519 T^{2} - 6459 T^{3} + 69550 T^{4} - 6459 p T^{5} + 519 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 19 T + 409 T^{2} + 4599 T^{3} + 55523 T^{4} + 4599 p T^{5} + 409 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 6 T + 331 T^{2} + 1665 T^{3} + 45369 T^{4} + 1665 p T^{5} + 331 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.73920393730311792437365132171, −5.64804604974124057112274309246, −5.30991127714986777303907536065, −5.13387473040994913831285510310, −4.90057305165983246131310722800, −4.67151223028101610510427589553, −4.53100469546927015169520071394, −4.48422581198254915971255010016, −4.44946657610292086779025271315, −4.13214814021546223900586261445, −3.68072885597457964720910009619, −3.64007059141125539874653707983, −3.53039131210019634430312099792, −3.32013826570425505374602754270, −3.11933053907910066528374418284, −2.82220375070645217073763709453, −2.75461920391329213199354242184, −2.37702974638330469847983146718, −2.33191786291916886142699181715, −2.13835886559120590038439609760, −2.09199631453111581184233084320, −1.69023455999897669829204606900, −1.44520716560148302961333834041, −1.33100727586941344482199163171, −0.882823393494967436227056349448, 0, 0, 0, 0, 0.882823393494967436227056349448, 1.33100727586941344482199163171, 1.44520716560148302961333834041, 1.69023455999897669829204606900, 2.09199631453111581184233084320, 2.13835886559120590038439609760, 2.33191786291916886142699181715, 2.37702974638330469847983146718, 2.75461920391329213199354242184, 2.82220375070645217073763709453, 3.11933053907910066528374418284, 3.32013826570425505374602754270, 3.53039131210019634430312099792, 3.64007059141125539874653707983, 3.68072885597457964720910009619, 4.13214814021546223900586261445, 4.44946657610292086779025271315, 4.48422581198254915971255010016, 4.53100469546927015169520071394, 4.67151223028101610510427589553, 4.90057305165983246131310722800, 5.13387473040994913831285510310, 5.30991127714986777303907536065, 5.64804604974124057112274309246, 5.73920393730311792437365132171

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