Properties

Label 8-2240e4-1.1-c1e4-0-10
Degree $8$
Conductor $2.518\times 10^{13}$
Sign $1$
Analytic cond. $102352.$
Root an. cond. $4.22924$
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  + 4·7-s + 6·9-s − 16·17-s − 2·25-s + 16·31-s − 16·41-s + 12·47-s + 10·49-s + 24·63-s + 16·71-s + 16·73-s + 16·79-s + 9·81-s + 24·89-s − 64·97-s + 4·103-s − 8·113-s − 64·119-s + 42·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 96·153-s + 157-s + ⋯
L(s)  = 1  + 1.51·7-s + 2·9-s − 3.88·17-s − 2/5·25-s + 2.87·31-s − 2.49·41-s + 1.75·47-s + 10/7·49-s + 3.02·63-s + 1.89·71-s + 1.87·73-s + 1.80·79-s + 81-s + 2.54·89-s − 6.49·97-s + 0.394·103-s − 0.752·113-s − 5.86·119-s + 3.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 7.76·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(102352.\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.120330714\)
\(L(\frac12)\) \(\approx\) \(5.120330714\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_1$ \( ( 1 - T )^{4} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \)
11$C_2^2$ \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 30 T^{2} + 179 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 92 T^{2} + 4086 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 180 T^{2} + 13526 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 140 T^{2} + 10134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 212 T^{2} + 18486 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 236 T^{2} + 22710 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 8 T + 146 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
79$D_{4}$ \( ( 1 - 8 T + 171 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 20 T^{2} + 6966 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 32 T + 447 T^{2} + 32 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.48991085905297438331229138434, −6.41123199012505121516792024927, −5.95144176814593440917898760768, −5.85688683730551518626843096004, −5.63020294566559132902020508002, −5.13194718548715802490333560504, −5.01695755611888114516715702097, −4.89515310649093579701171413291, −4.63117398905761460211304919075, −4.62004931163814357376851604413, −4.43516062559659256863195598025, −3.98099604121805060501299814626, −3.89614781438349224829597868743, −3.75386345304978171782149845304, −3.67682321486881283853588692878, −2.86135888640375600816115329601, −2.76346474051351013311277788204, −2.48896253181878358150426487108, −2.34687615563925598297920375453, −1.94213566134547477704475188163, −1.73390090913425593696017124909, −1.61926641243273537040398930121, −1.15973564745637400311973883715, −0.75709439623621517590616315508, −0.39939217807741391089895148624, 0.39939217807741391089895148624, 0.75709439623621517590616315508, 1.15973564745637400311973883715, 1.61926641243273537040398930121, 1.73390090913425593696017124909, 1.94213566134547477704475188163, 2.34687615563925598297920375453, 2.48896253181878358150426487108, 2.76346474051351013311277788204, 2.86135888640375600816115329601, 3.67682321486881283853588692878, 3.75386345304978171782149845304, 3.89614781438349224829597868743, 3.98099604121805060501299814626, 4.43516062559659256863195598025, 4.62004931163814357376851604413, 4.63117398905761460211304919075, 4.89515310649093579701171413291, 5.01695755611888114516715702097, 5.13194718548715802490333560504, 5.63020294566559132902020508002, 5.85688683730551518626843096004, 5.95144176814593440917898760768, 6.41123199012505121516792024927, 6.48991085905297438331229138434

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