Properties

Label 8-96e8-1.1-c1e4-0-9
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 $0$

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: induced by $\chi_{9216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{40} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.094826700\)
\(L(\frac12)\) \(\approx\) \(5.094826700\)
\(L(\frac{3}{2})\) 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

−5.44522249754678150352922396842, −5.06290772074344357694882618383, −5.05614398351850211144988490634, −4.71402316773439943883224182030, −4.69994492696638575884327230411, −4.44979713557172732262389466134, −4.32629267628356351082820007230, −4.06253047098692020774931688828, −4.01133987193256310485214361431, −3.66393692288977074652037825721, −3.58976477539694952324328336651, −3.49070344139592879068885906983, −3.38713744081949611526101569657, −2.84222635600986726552435268695, −2.72491448978443404533273645287, −2.63453388733317289073244939599, −2.41471782207082450786164746016, −1.81603226532849095313832267296, −1.75534487576057498731978088792, −1.66370985886183010971994515903, −1.44184299645144067216379426220, −1.08646826038270750513462293632, −0.804413752885254026448738341983, −0.47144492850248835439305255937, −0.34810070370882689301693943648, 0.34810070370882689301693943648, 0.47144492850248835439305255937, 0.804413752885254026448738341983, 1.08646826038270750513462293632, 1.44184299645144067216379426220, 1.66370985886183010971994515903, 1.75534487576057498731978088792, 1.81603226532849095313832267296, 2.41471782207082450786164746016, 2.63453388733317289073244939599, 2.72491448978443404533273645287, 2.84222635600986726552435268695, 3.38713744081949611526101569657, 3.49070344139592879068885906983, 3.58976477539694952324328336651, 3.66393692288977074652037825721, 4.01133987193256310485214361431, 4.06253047098692020774931688828, 4.32629267628356351082820007230, 4.44979713557172732262389466134, 4.69994492696638575884327230411, 4.71402316773439943883224182030, 5.05614398351850211144988490634, 5.06290772074344357694882618383, 5.44522249754678150352922396842

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