# Properties

 Label 4-103e2-1.1-c0e2-0-0 Degree $4$ Conductor $10609$ Sign $1$ Analytic cond. $0.00264233$ Root an. cond. $0.226723$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s − 7-s + 2·9-s − 13-s + 14-s − 17-s − 2·18-s − 19-s − 23-s + 2·25-s + 26-s − 29-s + 32-s + 34-s + 38-s − 41-s + 46-s − 2·50-s + 58-s − 59-s − 61-s − 2·63-s − 64-s − 79-s + 3·81-s + 82-s − 83-s + ⋯
 L(s)  = 1 − 2-s − 7-s + 2·9-s − 13-s + 14-s − 17-s − 2·18-s − 19-s − 23-s + 2·25-s + 26-s − 29-s + 32-s + 34-s + 38-s − 41-s + 46-s − 2·50-s + 58-s − 59-s − 61-s − 2·63-s − 64-s − 79-s + 3·81-s + 82-s − 83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 10609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 10609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$10609$$    =    $$103^{2}$$ Sign: $1$ Analytic conductor: $$0.00264233$$ Root analytic conductor: $$0.226723$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{103} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 10609,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1898228829$$ $$L(\frac12)$$ $$\approx$$ $$0.1898228829$$ $$L(1)$$ 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)$
bad103$C_1$ $$( 1 - T )^{2}$$
good2$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
3$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
5$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
7$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
11$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
13$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
17$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
19$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
23$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
29$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
31$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
37$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
41$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
43$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
47$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
53$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
59$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
61$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
67$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
73$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
79$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
83$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
89$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
97$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$