# Properties

 Label 4-360e2-1.1-c1e2-0-37 Degree $4$ Conductor $129600$ Sign $-1$ Analytic cond. $8.26340$ Root an. cond. $1.69546$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 8·13-s + 25-s − 8·37-s + 2·49-s − 20·61-s + 4·73-s − 20·97-s − 20·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
 L(s)  = 1 − 2.21·13-s + 1/5·25-s − 1.31·37-s + 2/7·49-s − 2.56·61-s + 0.468·73-s − 2.03·97-s − 1.91·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$129600$$    =    $$2^{6} \cdot 3^{4} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$8.26340$$ Root analytic conductor: $$1.69546$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 129600,\ (\ :1/2, 1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 $$1$$
3 $$1$$
5$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good7$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
43$C_2^2$ $$1 - 38 T^{2} + p^{2} T^{4}$$
47$C_2^2$ $$1 + 82 T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2^2$ $$1 + 106 T^{2} + p^{2} T^{4}$$
61$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
67$C_2^2$ $$1 - 86 T^{2} + p^{2} T^{4}$$
71$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
73$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
79$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
83$C_2^2$ $$1 + 58 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 + p T^{2} )^{2}$$
97$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$