# Properties

 Degree 4 Conductor $2^{2} \cdot 3^{2} \cdot 23^{2} \cdot 29^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 2·3-s + 3·4-s + 6·5-s + 4·6-s + 6·7-s + 4·8-s + 3·9-s + 12·10-s − 2·11-s + 6·12-s + 6·13-s + 12·14-s + 12·15-s + 5·16-s − 6·17-s + 6·18-s − 2·19-s + 18·20-s + 12·21-s − 4·22-s + 2·23-s + 8·24-s + 17·25-s + 12·26-s + 4·27-s + 18·28-s + ⋯
 L(s)  = 1 + 1.41·2-s + 1.15·3-s + 3/2·4-s + 2.68·5-s + 1.63·6-s + 2.26·7-s + 1.41·8-s + 9-s + 3.79·10-s − 0.603·11-s + 1.73·12-s + 1.66·13-s + 3.20·14-s + 3.09·15-s + 5/4·16-s − 1.45·17-s + 1.41·18-s − 0.458·19-s + 4.02·20-s + 2.61·21-s − 0.852·22-s + 0.417·23-s + 1.63·24-s + 17/5·25-s + 2.35·26-s + 0.769·27-s + 3.40·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 16016004 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 16016004 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$16016004$$    =    $$2^{2} \cdot 3^{2} \cdot 23^{2} \cdot 29^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{4002} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 16016004,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $32.13573044$ $L(\frac12)$ $\approx$ $32.13573044$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;23,\;29\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ $$( 1 - T )^{2}$$
3$C_1$ $$( 1 - T )^{2}$$
23$C_1$ $$( 1 - T )^{2}$$
29$C_1$ $$( 1 + T )^{2}$$
good5$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
7$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
11$D_{4}$ $$1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 - 6 T + 32 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 4 T + 51 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
41$C_2^2$ $$1 - 65 T^{2} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 14 T + 155 T^{2} - 14 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 10 T + 36 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 4 T + 170 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}