# Properties

 Degree 4 Conductor $2^{8} \cdot 3^{4} \cdot 7^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·3-s − 2·5-s − 7-s + 6·9-s + 11-s + 6·13-s + 6·15-s − 10·17-s + 14·19-s + 3·21-s + 4·23-s + 5·25-s − 9·27-s + 4·29-s − 6·31-s − 3·33-s + 2·35-s + 4·37-s − 18·39-s − 3·41-s − 43-s − 12·45-s + 30·51-s + 24·53-s − 2·55-s − 42·57-s − 7·59-s + ⋯
 L(s)  = 1 − 1.73·3-s − 0.894·5-s − 0.377·7-s + 2·9-s + 0.301·11-s + 1.66·13-s + 1.54·15-s − 2.42·17-s + 3.21·19-s + 0.654·21-s + 0.834·23-s + 25-s − 1.73·27-s + 0.742·29-s − 1.07·31-s − 0.522·33-s + 0.338·35-s + 0.657·37-s − 2.88·39-s − 0.468·41-s − 0.152·43-s − 1.78·45-s + 4.20·51-s + 3.29·53-s − 0.269·55-s − 5.56·57-s − 0.911·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$1016064$$    =    $$2^{8} \cdot 3^{4} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{1008} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 1016064,\ (\ :1/2, 1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$1.138363842$$ $$L(\frac12)$$ $$\approx$$ $$1.138363842$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3$C_2$ $$1 + p T + p T^{2}$$
7$C_2$ $$1 + T + T^{2}$$
good5$C_2^2$ $$1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4}$$
11$C_2^2$ $$1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4}$$
13$C_2^2$ $$1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
17$C_2$ $$( 1 + 5 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 7 T + p T^{2} )^{2}$$
23$C_2^2$ $$1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
31$C_2^2$ $$1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
37$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
41$C_2^2$ $$1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
43$C_2^2$ $$1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4}$$
47$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
59$C_2^2$ $$1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
67$C_2^2$ $$1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4}$$
71$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - T + p T^{2} )^{2}$$
79$C_2^2$ $$1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
83$C_2^2$ $$1 - 16 T + 173 T^{2} - 16 p T^{3} + p^{2} T^{4}$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}