Properties

Label 8-1792e4-1.1-c2e4-0-4
Degree $8$
Conductor $1.031\times 10^{13}$
Sign $1$
Analytic cond. $5.68449\times 10^{6}$
Root an. cond. $6.98773$
Motivic weight $2$
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  − 20·5-s + 20·9-s + 28·13-s − 8·17-s + 164·25-s − 104·29-s − 8·37-s − 8·41-s − 400·45-s − 14·49-s + 176·53-s + 60·61-s − 560·65-s + 104·73-s + 166·81-s + 160·85-s − 376·89-s + 152·97-s − 92·101-s − 136·109-s + 232·113-s + 560·117-s + 420·121-s − 420·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 4·5-s + 20/9·9-s + 2.15·13-s − 0.470·17-s + 6.55·25-s − 3.58·29-s − 0.216·37-s − 0.195·41-s − 8.88·45-s − 2/7·49-s + 3.32·53-s + 0.983·61-s − 8.61·65-s + 1.42·73-s + 2.04·81-s + 1.88·85-s − 4.22·89-s + 1.56·97-s − 0.910·101-s − 1.24·109-s + 2.05·113-s + 4.78·117-s + 3.47·121-s − 3.35·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(5.68449\times 10^{6}\)
Root analytic conductor: \(6.98773\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.024430558\)
\(L(\frac12)\) \(\approx\) \(2.024430558\)
\(L(2)\) 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 \( 1 \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 - 20 T^{2} + 26 p^{2} T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} \)
5$D_{4}$ \( ( 1 + 2 p T + 68 T^{2} + 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 420 T^{2} + 72934 T^{4} - 420 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 - 14 T + 380 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 4 T + 470 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 1092 T^{2} + 554026 T^{4} - 1092 p^{4} T^{6} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 1212 T^{2} + 919750 T^{4} - 1212 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 + 52 T + 1658 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 2916 T^{2} + 3944134 T^{4} - 2916 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 + 4 T + 2042 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 4 T + 566 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 2460 T^{2} + 7801702 T^{4} + 2460 p^{4} T^{6} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 - 3204 T^{2} + 4985734 T^{4} - 3204 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 - 88 T + 4754 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 3268 T^{2} + 16106730 T^{4} - 3268 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 - 30 T + 2564 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 4572 T^{2} + 42660838 T^{4} + 4572 p^{4} T^{6} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 15556 T^{2} + 106095174 T^{4} - 15556 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 52 T + 10326 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 19332 T^{2} + 170120326 T^{4} - 19332 p^{4} T^{6} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 12084 T^{2} + 87882154 T^{4} - 12084 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 + 188 T + 22886 T^{2} + 188 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 76 T + 4134 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−6.72581010251851886977589242387, −6.11833050830936658685088871040, −5.91160492037882563397315616149, −5.71949611427455716006494490413, −5.57872163617857905072989432482, −5.55262225106234633153071092214, −4.86804354488114254210026899700, −4.68049057477906085201623684639, −4.51308444305613233709667797675, −4.33050473569390314770111351655, −4.15711197586586106997943487345, −3.84041249829234508922313864804, −3.76687571790424362065528696960, −3.63799310304972030118127229691, −3.60754250955167345870127264588, −3.30621494575234509704325322670, −2.97809510741711930103276366513, −2.39331258945217503339920811763, −2.10270631330203062172651022347, −1.89928400577487200567174071332, −1.38740039245821302271736326058, −1.37270621507725520953623188288, −0.70775930265468900805309195254, −0.49290660169225735877906269972, −0.33582182751273260012874373758, 0.33582182751273260012874373758, 0.49290660169225735877906269972, 0.70775930265468900805309195254, 1.37270621507725520953623188288, 1.38740039245821302271736326058, 1.89928400577487200567174071332, 2.10270631330203062172651022347, 2.39331258945217503339920811763, 2.97809510741711930103276366513, 3.30621494575234509704325322670, 3.60754250955167345870127264588, 3.63799310304972030118127229691, 3.76687571790424362065528696960, 3.84041249829234508922313864804, 4.15711197586586106997943487345, 4.33050473569390314770111351655, 4.51308444305613233709667797675, 4.68049057477906085201623684639, 4.86804354488114254210026899700, 5.55262225106234633153071092214, 5.57872163617857905072989432482, 5.71949611427455716006494490413, 5.91160492037882563397315616149, 6.11833050830936658685088871040, 6.72581010251851886977589242387

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