Properties

Label 8-1856e4-1.1-c1e4-0-1
Degree $8$
Conductor $1.187\times 10^{13}$
Sign $1$
Analytic cond. $48241.4$
Root an. cond. $3.84970$
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  − 4·5-s + 10·9-s + 4·13-s − 10·25-s + 12·29-s − 40·45-s + 12·49-s + 36·53-s − 16·65-s + 57·81-s − 4·109-s + 40·117-s + 26·121-s + 80·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1.78·5-s + 10/3·9-s + 1.10·13-s − 2·25-s + 2.22·29-s − 5.96·45-s + 12/7·49-s + 4.94·53-s − 1.98·65-s + 19/3·81-s − 0.383·109-s + 3.69·117-s + 2.36·121-s + 7.15·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.23·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 29^{4}\)
Sign: $1$
Analytic conductor: \(48241.4\)
Root analytic conductor: \(3.84970\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 29^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.364846950\)
\(L(\frac12)\) \(\approx\) \(3.364846950\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
7$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 67 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.82303795133378861781760640969, −6.29313737708898311351266257793, −6.10442342874628735167884510683, −5.94321243355940964495909151327, −5.85934883382914454728278371033, −5.68546887437179798944110886366, −5.01548351931013075866145275791, −4.96119961592316698842263259113, −4.88977864695675656147302352774, −4.48599381485647143423684282332, −4.25410893852904764719896235974, −4.04455323334554107401268682186, −3.98569275150714746521046154814, −3.83964071288135945493792110630, −3.68805059734748792482479074305, −3.38028007260504647292946784243, −3.15058533608362532325851652377, −2.45821083886099028753469760210, −2.29563519122987647899978825162, −2.25753783679700072200181091523, −1.80645496015045977964939515932, −1.28999199747980159103584081060, −1.05557411044663441906650809178, −0.971158506903097112317827998623, −0.36670046549710525694892429205, 0.36670046549710525694892429205, 0.971158506903097112317827998623, 1.05557411044663441906650809178, 1.28999199747980159103584081060, 1.80645496015045977964939515932, 2.25753783679700072200181091523, 2.29563519122987647899978825162, 2.45821083886099028753469760210, 3.15058533608362532325851652377, 3.38028007260504647292946784243, 3.68805059734748792482479074305, 3.83964071288135945493792110630, 3.98569275150714746521046154814, 4.04455323334554107401268682186, 4.25410893852904764719896235974, 4.48599381485647143423684282332, 4.88977864695675656147302352774, 4.96119961592316698842263259113, 5.01548351931013075866145275791, 5.68546887437179798944110886366, 5.85934883382914454728278371033, 5.94321243355940964495909151327, 6.10442342874628735167884510683, 6.29313737708898311351266257793, 6.82303795133378861781760640969

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