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Properties

Label 8-2312e4-1.1-c0e4-0-0
Degree $8$
Conductor $2.857\times 10^{13}$
Sign $1$
Analytic cond. $1.77247$
Root an. cond. $1.07416$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 10·9-s − 16-s − 20·27-s + 4·41-s − 4·43-s + 4·48-s + 4·59-s − 4·73-s + 34·81-s − 4·83-s − 4·113-s − 2·121-s − 16·123-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + ⋯
L(s)  = 1  − 4·3-s + 10·9-s − 16-s − 20·27-s + 4·41-s − 4·43-s + 4·48-s + 4·59-s − 4·73-s + 34·81-s − 4·83-s − 4·113-s − 2·121-s − 16·123-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(1.77247\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 17^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06028044990\)
\(L(\frac12)\) \(\approx\) \(0.06028044990\)
\(L(1)\) 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)$
bad2$C_2^2$ \( 1 + T^{4} \)
17 \( 1 \)
good3$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
5$C_4\times C_2$ \( 1 + T^{8} \)
7$C_4\times C_2$ \( 1 + T^{8} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_4\times C_2$ \( 1 + T^{8} \)
31$C_4\times C_2$ \( 1 + T^{8} \)
37$C_4\times C_2$ \( 1 + T^{8} \)
41$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
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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

−6.53093375642412423827995305794, −6.41606049100249994401689411073, −6.23466122731871110397801416203, −5.75908470139773285013347766207, −5.71050147021768493865140506964, −5.64900797737924428192190244863, −5.49911660683255884088376966173, −5.13346346501427914224441101816, −5.01499858734307720280746666071, −4.89610692550134970422769839245, −4.38417976021814252730611519308, −4.35339893964438940697049711091, −4.31057017893872571455106530435, −4.05667128789788343886601695066, −3.76035386108548861258342091797, −3.59512313335099261093776754562, −3.14044592301142469205141518119, −2.65523861288052887151502368260, −2.43693930878925942531565385066, −2.29133800822006326032054308846, −1.61092849066624977269920437601, −1.51766173609441793194994682440, −1.19679698978785781686125433133, −1.04213239056380886333690434523, −0.18513576429461054648880901495, 0.18513576429461054648880901495, 1.04213239056380886333690434523, 1.19679698978785781686125433133, 1.51766173609441793194994682440, 1.61092849066624977269920437601, 2.29133800822006326032054308846, 2.43693930878925942531565385066, 2.65523861288052887151502368260, 3.14044592301142469205141518119, 3.59512313335099261093776754562, 3.76035386108548861258342091797, 4.05667128789788343886601695066, 4.31057017893872571455106530435, 4.35339893964438940697049711091, 4.38417976021814252730611519308, 4.89610692550134970422769839245, 5.01499858734307720280746666071, 5.13346346501427914224441101816, 5.49911660683255884088376966173, 5.64900797737924428192190244863, 5.71050147021768493865140506964, 5.75908470139773285013347766207, 6.23466122731871110397801416203, 6.41606049100249994401689411073, 6.53093375642412423827995305794

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