# Properties

 Label 4-315e2-1.1-c1e2-0-11 Degree $4$ Conductor $99225$ Sign $1$ Analytic cond. $6.32667$ Root an. cond. $1.58596$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 3·4-s + 5·16-s + 25-s + 20·37-s − 8·43-s − 7·49-s + 3·64-s + 24·67-s + 3·100-s − 28·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 60·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s − 24·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 + 3/2·4-s + 5/4·16-s + 1/5·25-s + 3.28·37-s − 1.21·43-s − 49-s + 3/8·64-s + 2.93·67-s + 3/10·100-s − 2.68·109-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.93·148-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 − 1.82·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$99225$$    =    $$3^{4} \cdot 5^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$6.32667$$ Root analytic conductor: $$1.58596$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{99225} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 99225,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.253375518$$ $$L(\frac12)$$ $$\approx$$ $$2.253375518$$ $$L(\frac{3}{2})$$ 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)$
bad3 $$1$$
5$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
7$C_2$ $$1 + p T^{2}$$
good2$C_2^2$ $$1 - 3 T^{2} + p^{2} T^{4}$$
11$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$C_2$ $$( 1 - p T^{2} )^{2}$$
29$C_2^2$ $$1 - 54 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
61$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
67$C_2$ $$( 1 - 12 T + p T^{2} )^{2}$$
71$C_2^2$ $$1 - 78 T^{2} + p^{2} T^{4}$$
73$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$