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Properties

Label 8-429e4-1.1-c0e4-0-1
Degree $8$
Conductor $33871089681$
Sign $1$
Analytic cond. $0.00210115$
Root an. cond. $0.462708$
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  + 3·2-s − 3-s + 6·4-s − 2·5-s − 3·6-s + 10·8-s − 6·10-s − 11-s − 6·12-s − 13-s + 2·15-s + 15·16-s − 12·20-s − 3·22-s − 10·24-s + 25-s − 3·26-s + 6·30-s + 22·32-s + 33-s + 39-s − 20·40-s − 2·41-s − 2·43-s − 6·44-s − 2·47-s − 15·48-s + ⋯
L(s)  = 1  + 3·2-s − 3-s + 6·4-s − 2·5-s − 3·6-s + 10·8-s − 6·10-s − 11-s − 6·12-s − 13-s + 2·15-s + 15·16-s − 12·20-s − 3·22-s − 10·24-s + 25-s − 3·26-s + 6·30-s + 22·32-s + 33-s + 39-s − 20·40-s − 2·41-s − 2·43-s − 6·44-s − 2·47-s − 15·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 11^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(0.00210115\)
Root analytic conductor: \(0.462708\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{429} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 11^{4} \cdot 13^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.547915163\)
\(L(\frac12)\) \(\approx\) \(1.547915163\)
\(L(1)\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
13$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good2$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
5$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
47$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
53$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
79$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
89$C_1$ \( ( 1 - T )^{8} \)
97$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−8.040127292498926733022239963097, −7.75808725216106026852526261124, −7.64481445688899107174014304403, −7.54582421722621227070330404234, −7.38570062039650306826004057530, −6.88297696684549833608361133753, −6.65313038405446641811858961413, −6.45505229696462178884846594522, −6.29271951071475403497349154053, −6.03462950567384111738172190494, −5.81474420226351671011310507734, −5.12491469255034135623183916362, −5.11629181579823547423922002679, −4.98623120538750706106171505842, −4.91429579640511628840861985634, −4.44456152672402843404268225822, −4.40620994771257051691464826572, −3.58722485375291194822162730118, −3.53248881238974829340002511082, −3.42983081131269022521373024226, −3.36516045675703410611326529525, −2.53536426066171617188298728752, −2.38355047805118523514237716335, −1.99052986627873227732121112811, −1.38013621677416743760713506671, 1.38013621677416743760713506671, 1.99052986627873227732121112811, 2.38355047805118523514237716335, 2.53536426066171617188298728752, 3.36516045675703410611326529525, 3.42983081131269022521373024226, 3.53248881238974829340002511082, 3.58722485375291194822162730118, 4.40620994771257051691464826572, 4.44456152672402843404268225822, 4.91429579640511628840861985634, 4.98623120538750706106171505842, 5.11629181579823547423922002679, 5.12491469255034135623183916362, 5.81474420226351671011310507734, 6.03462950567384111738172190494, 6.29271951071475403497349154053, 6.45505229696462178884846594522, 6.65313038405446641811858961413, 6.88297696684549833608361133753, 7.38570062039650306826004057530, 7.54582421722621227070330404234, 7.64481445688899107174014304403, 7.75808725216106026852526261124, 8.040127292498926733022239963097

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