# Properties

 Degree 4 Conductor $2^{2} \cdot 13^{2}$ 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 − 2·7-s − 4·8-s − 3·9-s + 12·10-s + 12·11-s + 6·12-s + 2·13-s + 4·14-s − 12·15-s + 5·16-s − 6·17-s + 6·18-s + 4·19-s − 18·20-s − 4·21-s − 24·22-s − 8·24-s + 17·25-s − 4·26-s − 14·27-s − 6·28-s + 12·29-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1.15·3-s + 3/2·4-s − 2.68·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s − 9-s + 3.79·10-s + 3.61·11-s + 1.73·12-s + 0.554·13-s + 1.06·14-s − 3.09·15-s + 5/4·16-s − 1.45·17-s + 1.41·18-s + 0.917·19-s − 4.02·20-s − 0.872·21-s − 5.11·22-s − 1.63·24-s + 17/5·25-s − 0.784·26-s − 2.69·27-s − 1.13·28-s + 2.22·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$676$$    =    $$2^{2} \cdot 13^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{676} (1, \cdot )$ Sato-Tate : $E_1$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 676,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.2658192833$ $L(\frac12)$ $\approx$ $0.2658192833$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;13\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$( 1 + T )^{2}$$
13$C_1$ $$( 1 - T )^{2}$$
good3$C_2$ $$( 1 - T + p T^{2} )^{2}$$
5$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
7$C_2$ $$( 1 + T + p T^{2} )^{2}$$
11$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
23$C_2$ $$( 1 + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 + p T^{2} )^{2}$$
43$C_2$ $$( 1 + T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
53$C_2$ $$( 1 + p T^{2} )^{2}$$
59$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 14 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
97$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}