Properties

Degree $8$
Conductor $6.154\times 10^{12}$
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  + 2·4-s − 2·9-s + 2·11-s + 16-s + 4·29-s − 4·36-s + 4·44-s + 49-s − 2·64-s − 4·71-s − 2·79-s + 3·81-s − 4·99-s + 4·109-s + 8·116-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 − 169-s + ⋯
L(s)  = 1  + 2·4-s − 2·9-s + 2·11-s + 16-s + 4·29-s − 4·36-s + 4·44-s + 49-s − 2·64-s − 4·71-s − 2·79-s + 3·81-s − 4·99-s + 4·109-s + 8·116-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 − 169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.063654534\)
\(L(\frac12)\) \(\approx\) \(2.063654534\)
\(L(1)\) 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)$
bad3$C_2$ \( ( 1 + T^{2} )^{2} \)
5 \( 1 \)
7$C_2^2$ \( 1 - T^{2} + T^{4} \)
good2$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
13$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
17$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
23$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{4} \)
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$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
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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

−7.04617732380829806092168981110, −6.51033319227279921106186720480, −6.37228678742432010156662628566, −6.36043914413746361929217544352, −6.30533529410754720474081293563, −5.79949499433153945011159035881, −5.79071754921210242733575081470, −5.77847644038558579992083573489, −5.19468295254110238710123648436, −4.82148606476732153881546481254, −4.72639573927937808472050199215, −4.67433901265314038003178145429, −4.14384563040902869233923008956, −4.02025632214192708359342399432, −3.88473803549106906982580107473, −3.22163759071610718594362779710, −3.01694497752613453450060376560, −2.98671574054719098737459256306, −2.89592097041120463039240858315, −2.57117304024433075536279854819, −2.05669390229901320017423026929, −1.98492320336306636557745181058, −1.65675978560010566305981127842, −1.07673924168719496661651400743, −0.933100740722115110091882544548, 0.933100740722115110091882544548, 1.07673924168719496661651400743, 1.65675978560010566305981127842, 1.98492320336306636557745181058, 2.05669390229901320017423026929, 2.57117304024433075536279854819, 2.89592097041120463039240858315, 2.98671574054719098737459256306, 3.01694497752613453450060376560, 3.22163759071610718594362779710, 3.88473803549106906982580107473, 4.02025632214192708359342399432, 4.14384563040902869233923008956, 4.67433901265314038003178145429, 4.72639573927937808472050199215, 4.82148606476732153881546481254, 5.19468295254110238710123648436, 5.77847644038558579992083573489, 5.79071754921210242733575081470, 5.79949499433153945011159035881, 6.30533529410754720474081293563, 6.36043914413746361929217544352, 6.37228678742432010156662628566, 6.51033319227279921106186720480, 7.04617732380829806092168981110

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