Properties

Label 8-1560e4-1.1-c1e4-0-4
Degree $8$
Conductor $5.922\times 10^{12}$
Sign $1$
Analytic cond. $24077.2$
Root an. cond. $3.52939$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 10·9-s + 4·13-s − 16·17-s − 24·23-s − 2·25-s + 20·27-s + 16·39-s + 4·49-s − 64·51-s + 8·53-s − 8·61-s − 96·69-s − 8·75-s + 16·79-s + 35·81-s + 16·101-s + 32·107-s + 48·113-s + 40·117-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 16·147-s + 149-s + ⋯
L(s)  = 1  + 2.30·3-s + 10/3·9-s + 1.10·13-s − 3.88·17-s − 5.00·23-s − 2/5·25-s + 3.84·27-s + 2.56·39-s + 4/7·49-s − 8.96·51-s + 1.09·53-s − 1.02·61-s − 11.5·69-s − 0.923·75-s + 1.80·79-s + 35/9·81-s + 1.59·101-s + 3.09·107-s + 4.51·113-s + 3.69·117-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·147-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(24077.2\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.890837459\)
\(L(\frac12)\) \(\approx\) \(6.890837459\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$D_{4}$ \( ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
59$D_4\times C_2$ \( 1 - 12 T^{2} - 5290 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 68 T^{2} + 10086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 188 T^{2} + 17766 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 164 T^{2} + 20742 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.82018985966045975947085867574, −6.44816651119498056662420047686, −6.24310100219035096770994133083, −6.22969186819600376702190917173, −5.92993244478442683364607437411, −5.76515490979629974613050217027, −5.68314064924324496929208136518, −4.92452454666002789685259393181, −4.74160069557908469244641392039, −4.61095698782783472847953192097, −4.47614689340581573276596584720, −4.11736279741253384691915458508, −3.97447528938125082979015326789, −3.85495911682734687008970311210, −3.58169674682013576088143160722, −3.32069045162436061317637858152, −3.11099499953696935587985133526, −2.59447098850014418079709855462, −2.39344231282427532720249758587, −2.12753298020841327654925831595, −1.89746195740558517590128789611, −1.81612965235776202488247120733, −1.78767231927762088122458355030, −0.63432609482743220931541321751, −0.49838258761453807251792423688, 0.49838258761453807251792423688, 0.63432609482743220931541321751, 1.78767231927762088122458355030, 1.81612965235776202488247120733, 1.89746195740558517590128789611, 2.12753298020841327654925831595, 2.39344231282427532720249758587, 2.59447098850014418079709855462, 3.11099499953696935587985133526, 3.32069045162436061317637858152, 3.58169674682013576088143160722, 3.85495911682734687008970311210, 3.97447528938125082979015326789, 4.11736279741253384691915458508, 4.47614689340581573276596584720, 4.61095698782783472847953192097, 4.74160069557908469244641392039, 4.92452454666002789685259393181, 5.68314064924324496929208136518, 5.76515490979629974613050217027, 5.92993244478442683364607437411, 6.22969186819600376702190917173, 6.24310100219035096770994133083, 6.44816651119498056662420047686, 6.82018985966045975947085867574

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