Properties

Label 8-9800e4-1.1-c1e4-0-6
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 9-s + 2·11-s + 10·13-s + 6·17-s + 4·23-s − 2·27-s − 2·29-s + 12·31-s + 4·33-s + 20·39-s − 12·41-s + 8·43-s − 2·47-s + 12·51-s + 4·53-s − 8·59-s − 20·61-s + 8·67-s + 8·69-s + 4·71-s + 16·73-s + 22·79-s − 5·81-s + 36·83-s − 4·87-s − 40·89-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/3·9-s + 0.603·11-s + 2.77·13-s + 1.45·17-s + 0.834·23-s − 0.384·27-s − 0.371·29-s + 2.15·31-s + 0.696·33-s + 3.20·39-s − 1.87·41-s + 1.21·43-s − 0.291·47-s + 1.68·51-s + 0.549·53-s − 1.04·59-s − 2.56·61-s + 0.977·67-s + 0.963·69-s + 0.474·71-s + 1.87·73-s + 2.47·79-s − 5/9·81-s + 3.95·83-s − 0.428·87-s − 4.23·89-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: induced by $\chi_{9800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(22.49803613\)
\(L(\frac12)\) \(\approx\) \(22.49803613\)
\(L(\frac{3}{2})\) 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$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 26 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 3 p T^{2} - 46 T^{3} + 480 T^{4} - 46 p T^{5} + 3 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 71 T^{2} - 2 p^{2} T^{3} + 1384 T^{4} - 2 p^{3} T^{5} + 71 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + p T^{2} - 62 T^{3} + 434 T^{4} - 62 p T^{5} + p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 50 T^{2} - 24 T^{3} + 1186 T^{4} - 24 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 54 T^{2} - 324 T^{3} + 1442 T^{4} - 324 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 73 T^{2} + 154 T^{3} + 2740 T^{4} + 154 p T^{5} + 73 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 134 T^{2} - 828 T^{3} + 5602 T^{4} - 828 p T^{5} + 134 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 106 T^{2} + 40 T^{3} + 5114 T^{4} + 40 p T^{5} + 106 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 68 T^{2} - 52 T^{3} - 2014 T^{4} - 52 p T^{5} + 68 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 124 T^{2} - 840 T^{3} + 7718 T^{4} - 840 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 147 T^{2} + 378 T^{3} + 9344 T^{4} + 378 p T^{5} + 147 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + 8866 T^{4} - 4 p^{2} T^{5} + 142 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 4 T + 72 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 288 T^{2} + 2572 T^{3} + 22350 T^{4} + 2572 p T^{5} + 288 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 154 T^{2} - 1176 T^{3} + 13994 T^{4} - 1176 p T^{5} + 154 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 32 T^{2} - 12 p T^{3} + 5438 T^{4} - 12 p^{2} T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 122 T^{2} + 240 T^{3} - 7038 T^{4} + 240 p T^{5} + 122 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 389 T^{2} - 4726 T^{3} + 48924 T^{4} - 4726 p T^{5} + 389 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 36 T + 750 T^{2} - 10564 T^{3} + 110978 T^{4} - 10564 p T^{5} + 750 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 40 T + 914 T^{2} + 13800 T^{3} + 152258 T^{4} + 13800 p T^{5} + 914 p^{2} T^{6} + 40 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 26 T + 545 T^{2} - 7602 T^{3} + 85890 T^{4} - 7602 p T^{5} + 545 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.36907785096475258308758835547, −5.24899129411892243793233264908, −5.04198056040237010434791307072, −4.79561806074447157559884750474, −4.58708868894098433818637247089, −4.37576923104059885573756165380, −4.33714600865827693222017693320, −3.97905034004753664663550584909, −3.73477665348731309023823373982, −3.56237860341802327964671384362, −3.49325794717638403380155091369, −3.44190203285991599167320536647, −3.21535085399720338908727440242, −2.83657717733648955849168638352, −2.76350737635017553950922573916, −2.61826122248111629899356513431, −2.49662122102152230702388645939, −1.96129716865278841929589310149, −1.76498340255698093387370757446, −1.54513486807675207950574556731, −1.51389083848461226381290509114, −1.07140873516174162550983641302, −0.879651669354760535680342041249, −0.60921113776433854256600946259, −0.47598627471980384270154156504, 0.47598627471980384270154156504, 0.60921113776433854256600946259, 0.879651669354760535680342041249, 1.07140873516174162550983641302, 1.51389083848461226381290509114, 1.54513486807675207950574556731, 1.76498340255698093387370757446, 1.96129716865278841929589310149, 2.49662122102152230702388645939, 2.61826122248111629899356513431, 2.76350737635017553950922573916, 2.83657717733648955849168638352, 3.21535085399720338908727440242, 3.44190203285991599167320536647, 3.49325794717638403380155091369, 3.56237860341802327964671384362, 3.73477665348731309023823373982, 3.97905034004753664663550584909, 4.33714600865827693222017693320, 4.37576923104059885573756165380, 4.58708868894098433818637247089, 4.79561806074447157559884750474, 5.04198056040237010434791307072, 5.24899129411892243793233264908, 5.36907785096475258308758835547

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