Properties

Label 8-1260e4-1.1-c1e4-0-8
Degree $8$
Conductor $2.520\times 10^{12}$
Sign $1$
Analytic cond. $10246.8$
Root an. cond. $3.17193$
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  − 2·5-s − 3·9-s − 4·11-s − 4·19-s + 5·25-s + 6·29-s + 24·31-s + 6·41-s + 6·45-s + 13·49-s + 8·55-s + 40·59-s − 8·61-s + 8·71-s − 40·79-s + 4·89-s + 8·95-s + 12·99-s − 2·101-s + 18·109-s + 26·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s − 12·145-s + ⋯
L(s)  = 1  − 0.894·5-s − 9-s − 1.20·11-s − 0.917·19-s + 25-s + 1.11·29-s + 4.31·31-s + 0.937·41-s + 0.894·45-s + 13/7·49-s + 1.07·55-s + 5.20·59-s − 1.02·61-s + 0.949·71-s − 4.50·79-s + 0.423·89-s + 0.820·95-s + 1.20·99-s − 0.199·101-s + 1.72·109-s + 2.36·121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.996·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(10246.8\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.223494620\)
\(L(\frac12)\) \(\approx\) \(2.223494620\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
good11$C_2^2$ \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
17$C_2^3$ \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \)
41$C_2^2$ \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 - 35 T^{2} - 624 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 93 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^3$ \( 1 + 6 T^{2} - 2773 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
83$C_2^3$ \( 1 + 165 T^{2} + 20336 T^{4} + 165 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^3$ \( 1 - 62 T^{2} - 5565 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.09462553806429372489403332895, −6.64134176078418437676488945741, −6.33223572283027153848711485724, −6.32532824614926095775415792719, −6.30643094190787590788549376790, −5.60955865325200803182940285162, −5.52504659447967438386676455652, −5.42390314237900186240675760550, −5.35831346695671842236570603437, −4.69755129049425450690784489283, −4.56773734912832350149798742708, −4.47977765494985742110457355739, −4.28807248974886227276154131672, −4.05656155200111800633372110465, −3.61423103965627699298796354721, −3.41502157076165248461461615003, −3.02756345812864052661285352268, −2.79950814976303339789848071524, −2.50831482135521642576806503797, −2.48226156604677491518912388100, −2.29880428324524935520961112979, −1.56188266929594408713622817517, −1.03307025629331222149879003405, −0.75625363033891706451668301275, −0.44321053662906926989720213235, 0.44321053662906926989720213235, 0.75625363033891706451668301275, 1.03307025629331222149879003405, 1.56188266929594408713622817517, 2.29880428324524935520961112979, 2.48226156604677491518912388100, 2.50831482135521642576806503797, 2.79950814976303339789848071524, 3.02756345812864052661285352268, 3.41502157076165248461461615003, 3.61423103965627699298796354721, 4.05656155200111800633372110465, 4.28807248974886227276154131672, 4.47977765494985742110457355739, 4.56773734912832350149798742708, 4.69755129049425450690784489283, 5.35831346695671842236570603437, 5.42390314237900186240675760550, 5.52504659447967438386676455652, 5.60955865325200803182940285162, 6.30643094190787590788549376790, 6.32532824614926095775415792719, 6.33223572283027153848711485724, 6.64134176078418437676488945741, 7.09462553806429372489403332895

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