Properties

Label 8-1400e4-1.1-c1e4-0-8
Degree $8$
Conductor $3.842\times 10^{12}$
Sign $1$
Analytic cond. $15617.8$
Root an. cond. $3.34350$
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  − 2·9-s − 8·11-s + 24·29-s − 18·31-s + 20·41-s + 13·49-s + 16·59-s − 16·61-s + 44·71-s + 18·79-s + 9·81-s + 22·89-s + 16·99-s + 16·101-s − 28·109-s + 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + ⋯
L(s)  = 1  − 2/3·9-s − 2.41·11-s + 4.45·29-s − 3.23·31-s + 3.12·41-s + 13/7·49-s + 2.08·59-s − 2.04·61-s + 5.22·71-s + 2.02·79-s + 81-s + 2.33·89-s + 1.60·99-s + 1.59·101-s − 2.68·109-s + 3.45·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(15617.8\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.763212673\)
\(L(\frac12)\) \(\approx\) \(2.763212673\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
good3$C_2^3$ \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 13 T^{2} - 2040 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 + 142 T^{2} + 14835 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 11 T + 32 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 73 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.78475213600872791645550723338, −6.61206436798506560101778272119, −6.53949024965363418464520995544, −6.10049209902807316810054933129, −6.02755175329970784288552027271, −5.48015728399402665245914785877, −5.45185554225357682759878629799, −5.30877296694420411482696149762, −5.30333182041235413725397517142, −4.81601773606973277345984539694, −4.57905428944298511994799032003, −4.52460607481560514787712500673, −4.15148386018535976327546402181, −3.71162870853228327437823158632, −3.54469443243529773928216285594, −3.48211116493203785859396962733, −3.02494918557174420545724658448, −2.61329960266129386410664422596, −2.43661604650621325620567295816, −2.37382794954065898906508732672, −2.27252238477387959755048433684, −1.66493926670771117910264701750, −0.942460150942411103579507833295, −0.798065529506043271744585590068, −0.44812490500234005034926079352, 0.44812490500234005034926079352, 0.798065529506043271744585590068, 0.942460150942411103579507833295, 1.66493926670771117910264701750, 2.27252238477387959755048433684, 2.37382794954065898906508732672, 2.43661604650621325620567295816, 2.61329960266129386410664422596, 3.02494918557174420545724658448, 3.48211116493203785859396962733, 3.54469443243529773928216285594, 3.71162870853228327437823158632, 4.15148386018535976327546402181, 4.52460607481560514787712500673, 4.57905428944298511994799032003, 4.81601773606973277345984539694, 5.30333182041235413725397517142, 5.30877296694420411482696149762, 5.45185554225357682759878629799, 5.48015728399402665245914785877, 6.02755175329970784288552027271, 6.10049209902807316810054933129, 6.53949024965363418464520995544, 6.61206436798506560101778272119, 6.78475213600872791645550723338

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