# Properties

 Label 2-140e2-1.1-c1-0-36 Degree $2$ Conductor $19600$ Sign $1$ Analytic cond. $156.506$ Root an. cond. $12.5102$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 3·9-s + 4·11-s + 2·13-s − 6·17-s + 8·19-s + 6·29-s + 8·31-s + 2·37-s − 2·41-s − 4·43-s + 8·47-s − 6·53-s + 6·61-s − 4·67-s + 8·71-s + 10·73-s − 16·79-s + 9·81-s − 8·83-s + 6·89-s − 6·97-s − 12·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
 L(s)  = 1 − 9-s + 1.20·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s + 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s − 0.824·53-s + 0.768·61-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 1.80·79-s + 81-s − 0.878·83-s + 0.635·89-s − 0.609·97-s − 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$19600$$    =    $$2^{4} \cdot 5^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$156.506$$ Root analytic conductor: $$12.5102$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{19600} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 19600,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.365210331$$ $$L(\frac12)$$ $$\approx$$ $$2.365210331$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
7 $$1$$
good3 $$1 + p T^{2}$$
11 $$1 - 4 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 - 8 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 - 8 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 + 2 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 - 8 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 6 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 - 10 T + p T^{2}$$
79 $$1 + 16 T + p T^{2}$$
83 $$1 + 8 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 + 6 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$