Properties

Label 8-75e4-1.1-c11e4-0-0
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $1.10272\times 10^{7}$
Root an. cond. $7.59116$
Motivic weight $11$
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  − 8.38e6·16-s − 1.18e9·31-s − 5.19e10·61-s − 3.13e10·81-s + 1.14e12·121-s + ⋯
L(s)  = 1  − 2·16-s − 7.43·31-s − 7.86·61-s − 81-s + 4·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.10272\times 10^{7}\)
Root analytic conductor: \(7.59116\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :11/2, 11/2, 11/2, 11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.02505891476\)
\(L(\frac12)\) \(\approx\) \(0.02505891476\)
\(L(\frac{13}{2})\) 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$C_2^2$ \( 1 + p^{22} T^{4} \)
5 \( 1 \)
good2$C_2$ \( ( 1 - p^{6} T + p^{11} T^{2} )^{2}( 1 + p^{6} T + p^{11} T^{2} )^{2} \)
7$C_2^3$ \( 1 - 3991163896037207977 T^{4} + p^{44} T^{8} \)
11$C_2$ \( ( 1 - p^{11} T^{2} )^{4} \)
13$C_2^3$ \( 1 - \)\(42\!\cdots\!37\)\( T^{4} + p^{44} T^{8} \)
17$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 190634130977363 T^{2} + p^{22} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p^{11} T^{2} )^{4} \)
31$C_2$ \( ( 1 + 296476943 T + p^{11} T^{2} )^{4} \)
37$C_2^3$ \( 1 + \)\(28\!\cdots\!38\)\( T^{4} + p^{44} T^{8} \)
41$C_2$ \( ( 1 - p^{11} T^{2} )^{4} \)
43$C_2^3$ \( 1 - \)\(11\!\cdots\!77\)\( T^{4} + p^{44} T^{8} \)
47$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p^{11} T^{2} )^{4} \)
61$C_2$ \( ( 1 + 12977292913 T + p^{11} T^{2} )^{4} \)
67$C_2^3$ \( 1 + \)\(13\!\cdots\!03\)\( T^{4} + p^{44} T^{8} \)
71$C_2$ \( ( 1 - p^{11} T^{2} )^{4} \)
73$C_2^3$ \( 1 - \)\(14\!\cdots\!42\)\( T^{4} + p^{44} T^{8} \)
79$C_2^2$ \( ( 1 - \)\(41\!\cdots\!42\)\( T^{2} + p^{22} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p^{11} T^{2} )^{4} \)
97$C_2^3$ \( 1 + \)\(29\!\cdots\!43\)\( T^{4} + p^{44} T^{8} \)
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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

−8.596025079438144577020363413166, −8.064949537659199132445725878084, −7.71010370403248707546899301642, −7.35819747138339932164560097592, −7.32486095404885809005014099712, −6.96278640965055431313490996628, −6.81627569277768886810951221110, −6.09474984546190884697324785816, −5.85876202139677678462217338241, −5.68736782786914442550082332390, −5.47632408374985773460074623581, −4.66430809191908726318449112666, −4.65286093876608140832538413653, −4.50315886648988168895441079517, −3.80367994291385343671087913458, −3.56210121892204335664908872914, −3.22870021691568552496723756488, −3.01541324804076927839993440163, −2.30976042954114051228697338909, −1.93091684941408812519159330772, −1.77216418948185215635305155796, −1.62489561597490699178297349571, −1.04400306494119350991663765681, −0.19980273231394446137887530405, −0.05279509049021667045244243105, 0.05279509049021667045244243105, 0.19980273231394446137887530405, 1.04400306494119350991663765681, 1.62489561597490699178297349571, 1.77216418948185215635305155796, 1.93091684941408812519159330772, 2.30976042954114051228697338909, 3.01541324804076927839993440163, 3.22870021691568552496723756488, 3.56210121892204335664908872914, 3.80367994291385343671087913458, 4.50315886648988168895441079517, 4.65286093876608140832538413653, 4.66430809191908726318449112666, 5.47632408374985773460074623581, 5.68736782786914442550082332390, 5.85876202139677678462217338241, 6.09474984546190884697324785816, 6.81627569277768886810951221110, 6.96278640965055431313490996628, 7.32486095404885809005014099712, 7.35819747138339932164560097592, 7.71010370403248707546899301642, 8.064949537659199132445725878084, 8.596025079438144577020363413166

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