# Properties

 Label 4-26e2-1.1-c1e2-0-2 Degree $4$ Conductor $676$ Sign $1$ Analytic cond. $0.0431023$ Root an. cond. $0.455643$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s + 4-s − 4·5-s + 9-s + 4·11-s − 2·12-s + 8·15-s + 16-s − 6·17-s + 8·19-s − 4·20-s − 4·23-s + 3·25-s − 2·27-s + 8·29-s − 8·33-s + 36-s − 4·37-s − 6·43-s + 4·44-s − 4·45-s + 16·47-s − 2·48-s − 13·49-s + 12·51-s + 12·53-s − 16·55-s + ⋯
 L(s)  = 1 − 1.15·3-s + 1/2·4-s − 1.78·5-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 2.06·15-s + 1/4·16-s − 1.45·17-s + 1.83·19-s − 0.894·20-s − 0.834·23-s + 3/5·25-s − 0.384·27-s + 1.48·29-s − 1.39·33-s + 1/6·36-s − 0.657·37-s − 0.914·43-s + 0.603·44-s − 0.596·45-s + 2.33·47-s − 0.288·48-s − 1.85·49-s + 1.68·51-s + 1.64·53-s − 2.15·55-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

 Degree: $$4$$ Conductor: $$676$$    =    $$2^{2} \cdot 13^{2}$$ Sign: $1$ Analytic conductor: $$0.0431023$$ Root analytic conductor: $$0.455643$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{676} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 676,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.3201552354$$ $$L(\frac12)$$ $$\approx$$ $$0.3201552354$$ $$L(\frac{3}{2})$$ 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)$
bad2$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
13$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good3$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + p T + p T^{2} )$$
5$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} )$$
7$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
11$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
19$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
23$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
29$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
31$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
41$C_2$ $$( 1 + p T^{2} )^{2}$$
43$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} )$$
47$C_2$$\times$$C_2$ $$( 1 - 13 T + p T^{2} )( 1 - 3 T + p T^{2} )$$
53$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + p T^{2} )$$
59$C_2$$\times$$C_2$ $$( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
61$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
71$C_2$$\times$$C_2$ $$( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
83$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + p T^{2} )$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$