Properties

Label 8-2160e4-1.1-c0e4-0-2
Degree $8$
Conductor $2.177\times 10^{13}$
Sign $1$
Analytic cond. $1.35034$
Root an. cond. $1.03825$
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  − 2·5-s + 25-s − 2·29-s + 2·41-s − 49-s − 2·61-s + 4·89-s + 4·101-s + 4·109-s − 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·5-s + 25-s − 2·29-s + 2·41-s − 49-s − 2·61-s + 4·89-s + 4·101-s + 4·109-s − 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1.35034\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6674095859\)
\(L(\frac12)\) \(\approx\) \(0.6674095859\)
\(L(1)\) 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 \( 1 \)
5$C_2$ \( ( 1 + T + T^{2} )^{2} \)
good7$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
11$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
89$C_2$ \( ( 1 - T + T^{2} )^{4} \)
97$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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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

−6.70061787975834571015686303652, −6.36389806479051389419857020897, −6.12640977067959197527110102804, −6.10152030812417905127427122256, −5.80882538162803068541835881723, −5.73698672350256743905285147228, −5.48617037245777466407456636448, −4.94316220882254008543057175965, −4.76348182084597636064235446332, −4.72765627252375934363096693239, −4.66258351014817769349426701055, −4.35368246118398338684239370523, −3.95832788026268393827972234958, −3.68691545613280419740794780920, −3.57165095179946575183583520492, −3.47721463976858957705235265307, −3.45474718625286887030640649981, −2.81541551353008979559568852167, −2.66039398739313116975375322202, −2.38439582429377656778591876945, −1.88191527200502079658863358174, −1.85507581649513027495424109356, −1.48127986216760581996466882832, −0.76524831502332005118360907554, −0.55141582021464210397343619677, 0.55141582021464210397343619677, 0.76524831502332005118360907554, 1.48127986216760581996466882832, 1.85507581649513027495424109356, 1.88191527200502079658863358174, 2.38439582429377656778591876945, 2.66039398739313116975375322202, 2.81541551353008979559568852167, 3.45474718625286887030640649981, 3.47721463976858957705235265307, 3.57165095179946575183583520492, 3.68691545613280419740794780920, 3.95832788026268393827972234958, 4.35368246118398338684239370523, 4.66258351014817769349426701055, 4.72765627252375934363096693239, 4.76348182084597636064235446332, 4.94316220882254008543057175965, 5.48617037245777466407456636448, 5.73698672350256743905285147228, 5.80882538162803068541835881723, 6.10152030812417905127427122256, 6.12640977067959197527110102804, 6.36389806479051389419857020897, 6.70061787975834571015686303652

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