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Properties

Label 8-3360e4-1.1-c0e4-0-7
Degree $8$
Conductor $1.275\times 10^{14}$
Sign $1$
Analytic cond. $7.90652$
Root an. cond. $1.29493$
Motivic weight $0$
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  + 4·23-s + 4·37-s − 81-s − 4·107-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 4·23-s + 4·37-s − 81-s − 4·107-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(7.90652\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.338941358\)
\(L(\frac12)\) \(\approx\) \(2.338941358\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
good11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + T^{4} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−6.35295122920641661242240613270, −6.12922659319171642054425028512, −5.87555554137069728187084209125, −5.63334491364519091240691944569, −5.46862014067402229555304726118, −5.18028170404715425964326431655, −5.15444193027023143104803891148, −5.07397893881095112942983844717, −4.47406885138366541483869650523, −4.44511167527059901729445315724, −4.29412392941798015255737200479, −4.10377789202258324898164171755, −4.09291264129803258365908783732, −3.29859349211270150538738217531, −3.26446227484495128333845277901, −3.25344723551540963051324191330, −2.82051512212679484165847567451, −2.74538573981288335019967272345, −2.62247524080756529876846945993, −2.03515831636750882701375497583, −1.99740591748223141957097562317, −1.56896874111692806635903628188, −1.01967869029384785975645187616, −1.01305516357556828690413184279, −0.75864946296297369517950204106, 0.75864946296297369517950204106, 1.01305516357556828690413184279, 1.01967869029384785975645187616, 1.56896874111692806635903628188, 1.99740591748223141957097562317, 2.03515831636750882701375497583, 2.62247524080756529876846945993, 2.74538573981288335019967272345, 2.82051512212679484165847567451, 3.25344723551540963051324191330, 3.26446227484495128333845277901, 3.29859349211270150538738217531, 4.09291264129803258365908783732, 4.10377789202258324898164171755, 4.29412392941798015255737200479, 4.44511167527059901729445315724, 4.47406885138366541483869650523, 5.07397893881095112942983844717, 5.15444193027023143104803891148, 5.18028170404715425964326431655, 5.46862014067402229555304726118, 5.63334491364519091240691944569, 5.87555554137069728187084209125, 6.12922659319171642054425028512, 6.35295122920641661242240613270

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