Properties

Label 8-252e4-1.1-c3e4-0-2
Degree $8$
Conductor $4032758016$
Sign $1$
Analytic cond. $48872.7$
Root an. cond. $3.85596$
Motivic weight $3$
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  + 14·7-s − 216·13-s − 4·19-s − 135·25-s − 402·31-s + 404·37-s − 976·43-s − 539·49-s + 1.17e3·61-s + 604·67-s + 892·73-s + 534·79-s − 3.02e3·91-s + 2.38e3·97-s − 3.32e3·103-s − 716·109-s + 2.27e3·121-s + 127-s + 131-s − 56·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.755·7-s − 4.60·13-s − 0.0482·19-s − 1.07·25-s − 2.32·31-s + 1.79·37-s − 3.46·43-s − 1.57·49-s + 2.46·61-s + 1.10·67-s + 1.43·73-s + 0.760·79-s − 3.48·91-s + 2.49·97-s − 3.18·103-s − 0.629·109-s + 1.71·121-s + 0.000698·127-s + 0.000666·131-s − 0.0365·133-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(48872.7\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.3183261570\)
\(L(\frac12)\) \(\approx\) \(0.3183261570\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_2$ \( ( 1 - p T + p^{3} T^{2} )^{2} \)
good5$C_2^3$ \( 1 + 27 p T^{2} + 104 p^{2} T^{4} + 27 p^{7} T^{6} + p^{12} T^{8} \)
11$C_2^3$ \( 1 - 207 p T^{2} + 28208 p^{2} T^{4} - 207 p^{7} T^{6} + p^{12} T^{8} \)
13$C_2$ \( ( 1 + 54 T + p^{3} T^{2} )^{4} \)
17$C_2^3$ \( 1 - 8286 T^{2} + 44520227 T^{4} - 8286 p^{6} T^{6} + p^{12} T^{8} \)
19$C_2^2$ \( ( 1 + 2 T - 6855 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 14166 T^{2} + 52639667 T^{4} + 14166 p^{6} T^{6} + p^{12} T^{8} \)
29$C_2^2$ \( ( 1 + 39153 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 201 T + 10610 T^{2} + 201 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 202 T - 9849 T^{2} - 202 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 83918 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 244 T + p^{3} T^{2} )^{4} \)
47$C_2^3$ \( 1 - 152206 T^{2} + 12387451107 T^{4} - 152206 p^{6} T^{6} + p^{12} T^{8} \)
53$C_2^3$ \( 1 + 121511 T^{2} - 7399438008 T^{4} + 121511 p^{6} T^{6} + p^{12} T^{8} \)
59$C_2^3$ \( 1 - 40773 T^{2} - 40518096112 T^{4} - 40773 p^{6} T^{6} + p^{12} T^{8} \)
61$C_2^2$ \( ( 1 - 588 T + 118763 T^{2} - 588 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 302 T - 209559 T^{2} - 302 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 691182 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 446 T - 190101 T^{2} - 446 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 267 T - 421750 T^{2} - 267 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 616509 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 1396078 T^{2} + 1452052491123 T^{4} - 1396078 p^{6} T^{6} + p^{12} T^{8} \)
97$C_2$ \( ( 1 - 595 T + p^{3} T^{2} )^{4} \)
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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

−8.306900718158293027444875763632, −7.981392442041803081205182117016, −7.890215065846854013450269710152, −7.47634216414866506911116059028, −7.16724015964520040040326211125, −7.13565736742979967032702202252, −6.83831574672218870126042385341, −6.67126008857050997191430789484, −5.99021531522195110303543430526, −5.91396919868294139377776527580, −5.46537440266212356541014304456, −5.07788522232706180584881873477, −4.86113864003008265101772054209, −4.82948380855040286715132314409, −4.75666685706617746851588198888, −3.97363654614933803426528433616, −3.71370592400233360931577904083, −3.45951102550312350940473824558, −2.85939335964358208518371418678, −2.46839515650623840412892435090, −2.22291068997121617426531803472, −1.95636050566667802057715614801, −1.57516009968359241277080048809, −0.62815749314091174127123133737, −0.13397200527784008886907156109, 0.13397200527784008886907156109, 0.62815749314091174127123133737, 1.57516009968359241277080048809, 1.95636050566667802057715614801, 2.22291068997121617426531803472, 2.46839515650623840412892435090, 2.85939335964358208518371418678, 3.45951102550312350940473824558, 3.71370592400233360931577904083, 3.97363654614933803426528433616, 4.75666685706617746851588198888, 4.82948380855040286715132314409, 4.86113864003008265101772054209, 5.07788522232706180584881873477, 5.46537440266212356541014304456, 5.91396919868294139377776527580, 5.99021531522195110303543430526, 6.67126008857050997191430789484, 6.83831574672218870126042385341, 7.13565736742979967032702202252, 7.16724015964520040040326211125, 7.47634216414866506911116059028, 7.890215065846854013450269710152, 7.981392442041803081205182117016, 8.306900718158293027444875763632

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