# Properties

 Label 4-295-1.1-c1e2-0-0 Degree $4$ Conductor $295$ Sign $1$ Analytic cond. $0.0188094$ Root an. cond. $0.370334$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s − 3-s + 4-s − 2·5-s + 2·6-s + 7-s + 9-s + 4·10-s + 2·11-s − 12-s − 2·13-s − 2·14-s + 2·15-s + 16-s − 2·18-s + 19-s − 2·20-s − 21-s − 4·22-s + 2·23-s − 2·25-s + 4·26-s − 4·27-s + 28-s − 9·29-s − 4·30-s + 4·31-s + ⋯
 L(s)  = 1 − 1.41·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 1.26·10-s + 0.603·11-s − 0.288·12-s − 0.554·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.471·18-s + 0.229·19-s − 0.447·20-s − 0.218·21-s − 0.852·22-s + 0.417·23-s − 2/5·25-s + 0.784·26-s − 0.769·27-s + 0.188·28-s − 1.67·29-s − 0.730·30-s + 0.718·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 295 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 295 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$295$$    =    $$5 \cdot 59$$ Sign: $1$ Analytic conductor: $$0.0188094$$ Root analytic conductor: $$0.370334$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 295,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.1492683688$$ $$L(\frac12)$$ $$\approx$$ $$0.1492683688$$ $$L(\frac{3}{2})$$ 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)$
bad5$C_1$$\times$$C_2$ $$( 1 + T )( 1 + T + p T^{2} )$$
59$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 4 T + p T^{2} )$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + p T + p T^{2} )$$
7$D_{4}$ $$1 - T - p T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
13$D_{4}$ $$1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
17$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
19$D_{4}$ $$1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 9 T + 64 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 - T + 68 T^{2} - p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 - 10 T + 90 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 9 T + 112 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 26 T^{2} + p^{2} T^{4}$$
67$C_2^2$ $$1 + 102 T^{2} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$C_2^2$ $$1 - 66 T^{2} + p^{2} T^{4}$$
79$D_{4}$ $$1 - T + 110 T^{2} - p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4}$$
89$C_2^2$ $$1 - 138 T^{2} + p^{2} T^{4}$$
97$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$