Properties

 Label 8-165e4-1.1-c2e4-0-2 Degree $8$ Conductor $741200625$ Sign $1$ Analytic cond. $408.578$ Root an. cond. $2.12035$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 − 12·3-s − 16·5-s + 90·9-s + 192·15-s + 23·16-s − 28·23-s + 142·25-s − 540·27-s + 80·31-s + 28·37-s − 1.44e3·45-s − 172·47-s − 276·48-s + 68·53-s + 88·59-s − 188·67-s + 336·69-s − 1.70e3·75-s − 368·80-s + 2.83e3·81-s + 400·89-s − 960·93-s + 172·97-s + 28·103-s − 336·111-s + 212·113-s + 448·115-s + ⋯
 L(s)  = 1 − 4·3-s − 3.19·5-s + 10·9-s + 64/5·15-s + 1.43·16-s − 1.21·23-s + 5.67·25-s − 20·27-s + 2.58·31-s + 0.756·37-s − 32·45-s − 3.65·47-s − 5.75·48-s + 1.28·53-s + 1.49·59-s − 2.80·67-s + 4.86·69-s − 22.7·75-s − 4.59·80-s + 35·81-s + 4.49·89-s − 10.3·93-s + 1.77·97-s + 0.271·103-s − 3.02·111-s + 1.87·113-s + 3.89·115-s + ⋯

Functional equation

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

Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 5^{4} \cdot 11^{4}$$ Sign: $1$ Analytic conductor: $$408.578$$ Root analytic conductor: $$2.12035$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 5^{4} \cdot 11^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.1380137496$$ $$L(\frac12)$$ $$\approx$$ $$0.1380137496$$ $$L(2)$$ 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$C_1$ $$( 1 + p T )^{4}$$
5$C_2$ $$( 1 + 8 T + p^{2} T^{2} )^{2}$$
11$C_2^2$ $$1 + 82 T^{2} + p^{4} T^{4}$$
good2$C_2^3$ $$1 - 23 T^{4} + p^{8} T^{8}$$
7$C_2^3$ $$1 + 1282 T^{4} + p^{8} T^{8}$$
13$C_2^3$ $$1 + 44002 T^{4} + p^{8} T^{8}$$
17$C_2^3$ $$1 - 5438 T^{4} + p^{8} T^{8}$$
19$C_2^2$ $$( 1 + 562 T^{2} + p^{4} T^{4} )^{2}$$
23$C_2^2$ $$( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 1322 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2$ $$( 1 - 20 T + p^{2} T^{2} )^{4}$$
37$C_2^2$ $$( 1 - 14 T + 98 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 1478 T^{2} + p^{4} T^{4} )^{2}$$
43$C_2^3$ $$1 + 549922 T^{4} + p^{8} T^{8}$$
47$C_2^2$ $$( 1 + 86 T + 3698 T^{2} + 86 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
53$C_2$ $$( 1 - 90 T + p^{2} T^{2} )^{2}( 1 + 56 T + p^{2} T^{2} )^{2}$$
59$C_2$ $$( 1 - 22 T + p^{2} T^{2} )^{4}$$
61$C_2^2$ $$( 1 + 1558 T^{2} + p^{4} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + 94 T + 4418 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 4318 T^{2} + p^{4} T^{4} )^{2}$$
73$C_2^3$ $$1 + 36867202 T^{4} + p^{8} T^{8}$$
79$C_2^2$ $$( 1 + 12442 T^{2} + p^{4} T^{4} )^{2}$$
83$C_2^3$ $$1 - 51909278 T^{4} + p^{8} T^{8}$$
89$C_2$ $$( 1 - 100 T + p^{2} T^{2} )^{4}$$
97$C_2^2$ $$( 1 - 86 T + 3698 T^{2} - 86 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$