Properties

Label 8-88e4-1.1-c0e4-0-0
Degree $8$
Conductor $59969536$
Sign $1$
Analytic cond. $3.72013\times 10^{-6}$
Root an. cond. $0.209565$
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  − 2-s − 2·3-s + 2·6-s + 9-s − 11-s − 2·17-s − 18-s + 3·19-s + 22-s − 25-s + 32-s + 2·33-s + 2·34-s − 3·38-s − 2·41-s − 2·43-s − 49-s + 50-s + 4·51-s − 6·57-s + 3·59-s − 64-s − 2·66-s − 2·67-s − 2·73-s + 2·75-s + 2·82-s + ⋯
L(s)  = 1  − 2-s − 2·3-s + 2·6-s + 9-s − 11-s − 2·17-s − 18-s + 3·19-s + 22-s − 25-s + 32-s + 2·33-s + 2·34-s − 3·38-s − 2·41-s − 2·43-s − 49-s + 50-s + 4·51-s − 6·57-s + 3·59-s − 64-s − 2·66-s − 2·67-s − 2·73-s + 2·75-s + 2·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59969536 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(59969536\)    =    \(2^{12} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(3.72013\times 10^{-6}\)
Root analytic conductor: \(0.209565\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 59969536,\ (\ :0, 0, 0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03037514870\)
\(L(\frac12)\) \(\approx\) \(0.03037514870\)
\(L(1)\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good3$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
5$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
7$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−11.07767912405533495553158013088, −10.25751161800844311628585574119, −10.15619435925391413893199930499, −10.13125239066839070428183229427, −9.801542251643198264879174078581, −9.284969942379507859608611312768, −9.191973812477788515768121620020, −8.700097417800058409783457980085, −8.433898624247411372751716098663, −8.341090551312655104382721172892, −7.66367238784448783056406444752, −7.48952021262817008111292203391, −7.28039020908791378434224755870, −6.66545754783887735413245583256, −6.38456810366459715900409676471, −6.29990722230209404329316790795, −5.50038095571044670886224694877, −5.47635468920967000080472360694, −5.32559889154914950171752856996, −4.65735988994615014311933168328, −4.63376475369904117861461445186, −3.67884878259526706903463378333, −3.29597448370317997586963941688, −2.68791914585162550379183914391, −1.80284319275365030447112589300, 1.80284319275365030447112589300, 2.68791914585162550379183914391, 3.29597448370317997586963941688, 3.67884878259526706903463378333, 4.63376475369904117861461445186, 4.65735988994615014311933168328, 5.32559889154914950171752856996, 5.47635468920967000080472360694, 5.50038095571044670886224694877, 6.29990722230209404329316790795, 6.38456810366459715900409676471, 6.66545754783887735413245583256, 7.28039020908791378434224755870, 7.48952021262817008111292203391, 7.66367238784448783056406444752, 8.341090551312655104382721172892, 8.433898624247411372751716098663, 8.700097417800058409783457980085, 9.191973812477788515768121620020, 9.284969942379507859608611312768, 9.801542251643198264879174078581, 10.13125239066839070428183229427, 10.15619435925391413893199930499, 10.25751161800844311628585574119, 11.07767912405533495553158013088

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