Properties

Label 8-1200e4-1.1-c1e4-0-1
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $8430.11$
Root an. cond. $3.09548$
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  − 12·13-s + 4·17-s − 20·37-s − 4·53-s + 16·61-s + 12·73-s − 81-s + 12·97-s + 72·101-s − 44·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 3.32·13-s + 0.970·17-s − 3.28·37-s − 0.549·53-s + 2.04·61-s + 1.40·73-s − 1/9·81-s + 1.21·97-s + 7.16·101-s − 4.13·113-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(8430.11\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1601650980\)
\(L(\frac12)\) \(\approx\) \(0.1601650980\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + T^{4} \)
5 \( 1 \)
good7$C_2^3$ \( 1 - 94 T^{4} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 158 T^{4} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2^3$ \( 1 + 1202 T^{4} + p^{4} T^{8} \)
47$C_2^3$ \( 1 + 1666 T^{4} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
67$C_2^3$ \( 1 + 4946 T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 8722 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.01077617031745985239779039422, −6.80314370290963417859962625868, −6.64987701930415734535918200463, −6.20250003676782334101399318827, −6.08245503354473245912528622181, −6.05530462323691858004681362040, −5.39516422145205998460850088427, −5.26328172477102018476667521457, −5.15465300860496654286150259793, −4.89674405062634080836484397402, −4.85959361786151824557652541883, −4.79642445434659936435259054665, −3.99654835336251417459405434039, −3.92988506579200769464833640712, −3.82695376758861611073125842380, −3.43365806537013566984352240743, −3.07801849141329416927824617837, −2.95930554230974642069094052993, −2.54464444154210464650021074120, −2.16989105614327311718151528191, −2.14363232076853599274413629038, −1.82988116244737665243717049320, −1.23704812270115128389863172673, −0.836640434130778066386649387967, −0.095019547456513772752380996130, 0.095019547456513772752380996130, 0.836640434130778066386649387967, 1.23704812270115128389863172673, 1.82988116244737665243717049320, 2.14363232076853599274413629038, 2.16989105614327311718151528191, 2.54464444154210464650021074120, 2.95930554230974642069094052993, 3.07801849141329416927824617837, 3.43365806537013566984352240743, 3.82695376758861611073125842380, 3.92988506579200769464833640712, 3.99654835336251417459405434039, 4.79642445434659936435259054665, 4.85959361786151824557652541883, 4.89674405062634080836484397402, 5.15465300860496654286150259793, 5.26328172477102018476667521457, 5.39516422145205998460850088427, 6.05530462323691858004681362040, 6.08245503354473245912528622181, 6.20250003676782334101399318827, 6.64987701930415734535918200463, 6.80314370290963417859962625868, 7.01077617031745985239779039422

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