Properties

Label 8-3920e4-1.1-c1e4-0-1
Degree $8$
Conductor $2.361\times 10^{14}$
Sign $1$
Analytic cond. $959959.$
Root an. cond. $5.59476$
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  + 2·3-s − 4·5-s − 9-s − 2·11-s − 10·13-s − 8·15-s − 6·17-s + 4·23-s + 10·25-s − 2·27-s − 2·29-s − 12·31-s − 4·33-s − 20·39-s − 12·41-s + 8·43-s + 4·45-s − 2·47-s − 12·51-s − 4·53-s + 8·55-s + 8·59-s − 20·61-s + 40·65-s + 8·67-s + 8·69-s − 4·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 1/3·9-s − 0.603·11-s − 2.77·13-s − 2.06·15-s − 1.45·17-s + 0.834·23-s + 2·25-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 + 0.596·45-s − 0.291·47-s − 1.68·51-s − 0.549·53-s + 1.07·55-s + 1.04·59-s − 2.56·61-s + 4.96·65-s + 0.977·67-s + 0.963·69-s − 0.474·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(959959.\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 + T )^{4} \)
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

−6.59050551580783476975298191236, −6.00527767061376915983765883787, −5.99320902005876404141593064329, −5.70451326424039637634656107966, −5.60327907018635049002478990945, −5.26702953586462059524889202200, −5.16169244680729056450784446785, −4.92620362068911681919703936604, −4.69501658284931436752912629323, −4.56838597758166729612011108970, −4.36627858714822039759696287833, −4.17398054714658365197633070452, −4.09449635494322505409505776729, −3.58839845121882205371791950649, −3.38144591894135661928402628872, −3.36251426441858865369844427257, −3.13876668177928411668777564603, −2.77663185061918619625674330703, −2.61114022279491108484573982544, −2.46014252882774739442021277849, −2.37212688521159150142968489047, −1.97413424286168116531305493673, −1.67707080723812019486224651713, −1.19111182230048337096050547225, −1.14249421162625901878574292394, 0, 0, 0, 0, 1.14249421162625901878574292394, 1.19111182230048337096050547225, 1.67707080723812019486224651713, 1.97413424286168116531305493673, 2.37212688521159150142968489047, 2.46014252882774739442021277849, 2.61114022279491108484573982544, 2.77663185061918619625674330703, 3.13876668177928411668777564603, 3.36251426441858865369844427257, 3.38144591894135661928402628872, 3.58839845121882205371791950649, 4.09449635494322505409505776729, 4.17398054714658365197633070452, 4.36627858714822039759696287833, 4.56838597758166729612011108970, 4.69501658284931436752912629323, 4.92620362068911681919703936604, 5.16169244680729056450784446785, 5.26702953586462059524889202200, 5.60327907018635049002478990945, 5.70451326424039637634656107966, 5.99320902005876404141593064329, 6.00527767061376915983765883787, 6.59050551580783476975298191236

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