Properties

Label 8-882e4-1.1-c2e4-0-12
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $333591.$
Root an. cond. $4.90232$
Motivic weight $2$
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  + 4·4-s + 12·16-s + 52·25-s − 188·37-s + 124·43-s + 32·64-s − 124·67-s + 164·79-s + 208·100-s + 676·109-s + 92·121-s + 127-s + 131-s + 137-s + 139-s − 752·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 670·169-s + 496·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 4-s + 3/4·16-s + 2.07·25-s − 5.08·37-s + 2.88·43-s + 1/2·64-s − 1.85·67-s + 2.07·79-s + 2.07·100-s + 6.20·109-s + 0.760·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 5.08·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.96·169-s + 2.88·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(333591.\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.760539712\)
\(L(\frac12)\) \(\approx\) \(5.760539712\)
\(L(2)\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 - 26 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 335 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 554 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 145 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 986 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 1775 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 47 T + p^{2} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 1342 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 31 T + p^{2} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 2518 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 214 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 26 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 530 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 31 T + p^{2} T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 6554 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 4031 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 41 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 13754 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 12386 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 17090 T^{2} + p^{4} T^{4} )^{2} \)
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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

−7.08591055762002227522796539815, −6.83549312817224574484701577047, −6.74483563311812694998883587945, −6.49878876514486905018689092524, −6.06642268076916765132650969572, −5.82935132901021588695285958400, −5.81677267436695032583240872283, −5.56502647622743463685627132144, −5.09567454881088082301706078641, −5.08900939356351530314438707525, −4.60781146643723200598155846865, −4.46050875014091990241514592414, −4.44050611416465110061971702525, −3.77923312379817799124356932150, −3.42516704491173606144146485569, −3.40264655621862162388385200684, −3.12119082862025654780059277589, −2.98512590766695091020773263592, −2.42124336308246298821648205328, −2.03953776651752591968572251027, −1.91609415668378748187104919670, −1.75466802421105957384220217213, −1.00679076852491754895756017456, −0.837678439018601728425428380588, −0.37392850919022584629115410885, 0.37392850919022584629115410885, 0.837678439018601728425428380588, 1.00679076852491754895756017456, 1.75466802421105957384220217213, 1.91609415668378748187104919670, 2.03953776651752591968572251027, 2.42124336308246298821648205328, 2.98512590766695091020773263592, 3.12119082862025654780059277589, 3.40264655621862162388385200684, 3.42516704491173606144146485569, 3.77923312379817799124356932150, 4.44050611416465110061971702525, 4.46050875014091990241514592414, 4.60781146643723200598155846865, 5.08900939356351530314438707525, 5.09567454881088082301706078641, 5.56502647622743463685627132144, 5.81677267436695032583240872283, 5.82935132901021588695285958400, 6.06642268076916765132650969572, 6.49878876514486905018689092524, 6.74483563311812694998883587945, 6.83549312817224574484701577047, 7.08591055762002227522796539815

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