Properties

 Degree 8 Conductor $5^{4} \cdot 11^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

Origins of factors

Dirichlet series

 L(s)  = 1 − 2·3-s + 2·9-s − 8·16-s + 18·23-s + 25-s + 4·27-s − 14·37-s − 24·47-s + 16·48-s − 12·53-s + 26·67-s − 36·69-s − 12·71-s − 2·75-s − 3·81-s − 34·97-s + 8·103-s + 28·111-s − 42·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 48·141-s − 16·144-s + 149-s + ⋯
 L(s)  = 1 − 1.15·3-s + 2/3·9-s − 2·16-s + 3.75·23-s + 1/5·25-s + 0.769·27-s − 2.30·37-s − 3.50·47-s + 2.30·48-s − 1.64·53-s + 3.17·67-s − 4.33·69-s − 1.42·71-s − 0.230·75-s − 1/3·81-s − 3.45·97-s + 0.788·103-s + 2.65·111-s − 3.95·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.04·141-s − 4/3·144-s + 0.0819·149-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$8$$ $$N$$ = $$9150625$$    =    $$5^{4} \cdot 11^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{55} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 9150625,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.386846$ $L(\frac12)$ $\approx$ $0.386846$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;11\}$, $$F_p(T)$$ is a polynomial of degree 8. If $p \in \{5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_2^2$ $$1 - T^{2} + p^{2} T^{4}$$
11$C_2$ $$( 1 - p T^{2} )^{2}$$
good2$C_2$ $$( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2}$$
3$C_2$$\times$$C_2^2$ $$( 1 + T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} )$$
7$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
19$C_2$ $$( 1 + p T^{2} )^{4}$$
23$C_2$$\times$$C_2^2$ $$( 1 - 9 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} )$$
29$C_2$ $$( 1 + p T^{2} )^{4}$$
31$C_2^2$ $$( 1 - 37 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2$$\times$$C_2^2$ $$( 1 + 7 T + p T^{2} )^{2}( 1 - 25 T^{2} + p^{2} T^{4} )$$
41$C_2$ $$( 1 - p T^{2} )^{4}$$
43$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
47$C_2$$\times$$C_2^2$ $$( 1 + 12 T + p T^{2} )^{2}( 1 + 50 T^{2} + p^{2} T^{4} )$$
53$C_2$$\times$$C_2^2$ $$( 1 + 6 T + p T^{2} )^{2}( 1 - 70 T^{2} + p^{2} T^{4} )$$
59$C_2$ $$( 1 - 15 T + p T^{2} )^{2}( 1 + 15 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 - p T^{2} )^{4}$$
67$C_2$$\times$$C_2^2$ $$( 1 - 13 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} )$$
71$C_2$ $$( 1 + 3 T + p T^{2} )^{4}$$
73$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
79$C_2$ $$( 1 + p T^{2} )^{4}$$
83$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 97 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2$$\times$$C_2^2$ $$( 1 + 17 T + p T^{2} )^{2}( 1 + 95 T^{2} + p^{2} T^{4} )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}