Properties

Label 8-825e4-1.1-c2e4-0-0
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $255361.$
Root an. cond. $4.74126$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 6·9-s + 20·11-s − 17·16-s + 92·23-s + 80·31-s − 12·36-s + 64·37-s − 40·44-s − 100·47-s + 148·49-s − 28·53-s + 40·59-s + 44·64-s + 136·67-s − 284·71-s + 27·81-s + 304·89-s − 184·92-s + 376·97-s + 120·99-s − 272·103-s − 208·113-s + 70·121-s − 160·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 2/3·9-s + 1.81·11-s − 1.06·16-s + 4·23-s + 2.58·31-s − 1/3·36-s + 1.72·37-s − 0.909·44-s − 2.12·47-s + 3.02·49-s − 0.528·53-s + 0.677·59-s + 0.687·64-s + 2.02·67-s − 4·71-s + 1/3·81-s + 3.41·89-s − 2·92-s + 3.87·97-s + 1.21·99-s − 2.64·103-s − 1.84·113-s + 0.578·121-s − 1.29·124-s + 0.00787·127-s + 0.00763·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(255361.\)
Root analytic conductor: \(4.74126\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(8.790368307\)
\(L(\frac12)\) \(\approx\) \(8.790368307\)
\(L(2)\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
5 \( 1 \)
11$D_{4}$ \( 1 - 20 T + 30 p T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} \)
good2$C_2^2 \wr C_2$ \( 1 + p T^{2} + 21 T^{4} + p^{5} T^{6} + p^{8} T^{8} \)
7$C_2^2 \wr C_2$ \( 1 - 148 T^{2} + 9978 T^{4} - 148 p^{4} T^{6} + p^{8} T^{8} \)
13$C_2^2 \wr C_2$ \( 1 - 4 p T^{2} - 6150 T^{4} - 4 p^{5} T^{6} + p^{8} T^{8} \)
17$C_2^2 \wr C_2$ \( 1 - 940 T^{2} + 386214 T^{4} - 940 p^{4} T^{6} + p^{8} T^{8} \)
19$C_2^2 \wr C_2$ \( 1 - 508 T^{2} + 116070 T^{4} - 508 p^{4} T^{6} + p^{8} T^{8} \)
23$D_{4}$ \( ( 1 - 2 p T + 1512 T^{2} - 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \)
29$C_2^2 \wr C_2$ \( 1 - 1828 T^{2} + 1730790 T^{4} - 1828 p^{4} T^{6} + p^{8} T^{8} \)
31$D_{4}$ \( ( 1 - 40 T + 1350 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 32 T + 2982 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - 4420 T^{2} + 9212934 T^{4} - 4420 p^{4} T^{6} + p^{8} T^{8} \)
43$C_2^2 \wr C_2$ \( 1 - 124 T^{2} - 2536026 T^{4} - 124 p^{4} T^{6} + p^{8} T^{8} \)
47$D_{4}$ \( ( 1 + 50 T + 5016 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 14 T + 5304 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 - 20 T + 30 p T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
61$C_2^2 \wr C_2$ \( 1 - 8740 T^{2} + 39948282 T^{4} - 8740 p^{4} T^{6} + p^{8} T^{8} \)
67$C_2$ \( ( 1 - 34 T + p^{2} T^{2} )^{4} \)
71$D_{4}$ \( ( 1 + 2 p T + 15120 T^{2} + 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \)
73$C_2^2 \wr C_2$ \( 1 - 16708 T^{2} + 126086406 T^{4} - 16708 p^{4} T^{6} + p^{8} T^{8} \)
79$C_2^2 \wr C_2$ \( 1 - 14836 T^{2} + 112926714 T^{4} - 14836 p^{4} T^{6} + p^{8} T^{8} \)
83$C_2^2 \wr C_2$ \( 1 - 22372 T^{2} + 213329190 T^{4} - 22372 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 - 152 T + 20646 T^{2} - 152 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 188 T + 22362 T^{2} - 188 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−7.01523067728902575705910874309, −6.82893813748519801150926799723, −6.73451365392434235077251288241, −6.54186396194847855162332828109, −6.19449687919121412434771481157, −6.18190055473715647026872207752, −5.81735186363690698635148936785, −5.29401565688610845401670473561, −5.20994027576057035557404116699, −4.92976645678553326439418435172, −4.76265353752731449974589689272, −4.50755893619859906816796994460, −4.16758637263169974389648538989, −4.06012811316214124312160230884, −3.97857235517716099384463709508, −3.30543713154342179500215883666, −3.08656092469767234555083715511, −2.85598031219343364047618055633, −2.79529343746089224906864336311, −2.05210251186953868394301412097, −1.98161192832399728035283220736, −1.35755137809531496319520345608, −1.02946983206191163177223232671, −0.75520100956935371851162298349, −0.62095999643772761487767095026, 0.62095999643772761487767095026, 0.75520100956935371851162298349, 1.02946983206191163177223232671, 1.35755137809531496319520345608, 1.98161192832399728035283220736, 2.05210251186953868394301412097, 2.79529343746089224906864336311, 2.85598031219343364047618055633, 3.08656092469767234555083715511, 3.30543713154342179500215883666, 3.97857235517716099384463709508, 4.06012811316214124312160230884, 4.16758637263169974389648538989, 4.50755893619859906816796994460, 4.76265353752731449974589689272, 4.92976645678553326439418435172, 5.20994027576057035557404116699, 5.29401565688610845401670473561, 5.81735186363690698635148936785, 6.18190055473715647026872207752, 6.19449687919121412434771481157, 6.54186396194847855162332828109, 6.73451365392434235077251288241, 6.82893813748519801150926799723, 7.01523067728902575705910874309

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