# Properties

 Label 4-1088-1.1-c1e2-0-0 Degree $4$ Conductor $1088$ Sign $1$ Analytic cond. $0.0693718$ Root an. cond. $0.513210$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

This L-function arises from both an elliptic curve over a number field, and from a genus 2 curve over the rationals. It is probably the L-function of smallest conductor with that property.

## Dirichlet series

 L(s)  = 1 − 2·4-s − 2·7-s − 2·9-s + 4·16-s + 5·17-s − 6·23-s + 2·25-s + 4·28-s − 2·31-s + 4·36-s + 12·47-s − 2·49-s + 4·63-s − 8·64-s − 10·68-s + 6·71-s − 20·73-s − 2·79-s − 5·81-s + 12·92-s − 8·97-s − 4·100-s − 8·103-s − 8·112-s + 24·113-s − 10·119-s − 10·121-s + ⋯
 L(s)  = 1 − 4-s − 0.755·7-s − 2/3·9-s + 16-s + 1.21·17-s − 1.25·23-s + 2/5·25-s + 0.755·28-s − 0.359·31-s + 2/3·36-s + 1.75·47-s − 2/7·49-s + 0.503·63-s − 64-s − 1.21·68-s + 0.712·71-s − 2.34·73-s − 0.225·79-s − 5/9·81-s + 1.25·92-s − 0.812·97-s − 2/5·100-s − 0.788·103-s − 0.755·112-s + 2.25·113-s − 0.916·119-s − 0.909·121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1088$$    =    $$2^{6} \cdot 17$$ Sign: $1$ Analytic conductor: $$0.0693718$$ Root analytic conductor: $$0.513210$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{1088} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1088,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4366701692$$ $$L(\frac12)$$ $$\approx$$ $$0.4366701692$$ $$L(\frac{3}{2})$$ 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$C_2$ $$1 + p T^{2}$$
17$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 6 T + p T^{2} )$$
good3$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
5$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
19$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
29$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
41$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2^2$ $$1 + 58 T^{2} + p^{2} T^{4}$$
47$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
53$C_2^2$ $$1 - 26 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
67$C_2^2$ $$1 - 26 T^{2} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
73$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
79$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
83$C_2^2$ $$1 - 86 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$