Properties

Degree $8$
Conductor $2.252\times 10^{12}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·9-s + 2·11-s + 16-s + 4·29-s − 2·36-s − 2·44-s − 2·64-s − 4·71-s − 2·79-s + 81-s + 4·99-s − 2·109-s − 4·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + ⋯
L(s)  = 1  − 4-s + 2·9-s + 2·11-s + 16-s + 4·29-s − 2·36-s − 2·44-s − 2·64-s − 4·71-s − 2·79-s + 81-s + 4·99-s − 2·109-s − 4·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{1225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.267105257\)
\(L(\frac12)\) \(\approx\) \(1.267105257\)
\(L(1)\) 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)$
bad5 \( 1 \)
7 \( 1 \)
good2$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
3$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
29$C_2$ \( ( 1 - T + T^{2} )^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
71$C_2$ \( ( 1 + T + T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{4} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−7.11801742361243060241271858594, −6.94733325653433445564612249417, −6.72996770311290153799569698002, −6.48433447341575731455902893978, −6.37205653261201891536893642438, −6.15832901537976846558656363582, −5.84666717386432890729971227593, −5.56296872112346410891689121620, −5.50593620852497924811778099128, −4.95337304366455609734181093077, −4.62448274509125449747838576721, −4.61555643021546928265907994248, −4.54118539395253706201611134673, −4.26322602047823058519856745019, −4.00195891924855911588036356450, −3.85225095112372907557771622793, −3.53465378423429131960271617445, −3.03678123928940308422767922130, −2.89538604763926493334569731822, −2.85617895297746116276147883246, −2.18037727613486672710851906101, −1.76351664832382944833496101964, −1.34952014963397499355087515702, −1.19080637093051140081446699663, −1.07470219746686778742392255655, 1.07470219746686778742392255655, 1.19080637093051140081446699663, 1.34952014963397499355087515702, 1.76351664832382944833496101964, 2.18037727613486672710851906101, 2.85617895297746116276147883246, 2.89538604763926493334569731822, 3.03678123928940308422767922130, 3.53465378423429131960271617445, 3.85225095112372907557771622793, 4.00195891924855911588036356450, 4.26322602047823058519856745019, 4.54118539395253706201611134673, 4.61555643021546928265907994248, 4.62448274509125449747838576721, 4.95337304366455609734181093077, 5.50593620852497924811778099128, 5.56296872112346410891689121620, 5.84666717386432890729971227593, 6.15832901537976846558656363582, 6.37205653261201891536893642438, 6.48433447341575731455902893978, 6.72996770311290153799569698002, 6.94733325653433445564612249417, 7.11801742361243060241271858594

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