# Properties

 Label 2-40e2-1.1-c3-0-103 Degree $2$ Conductor $1600$ Sign $-1$ Analytic cond. $94.4030$ Root an. cond. $9.71612$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 7·3-s − 6·7-s + 22·9-s + 43·11-s − 28·13-s − 91·17-s + 35·19-s − 42·21-s − 162·23-s − 35·27-s − 160·29-s + 42·31-s + 301·33-s − 314·37-s − 196·39-s − 203·41-s + 92·43-s − 196·47-s − 307·49-s − 637·51-s + 82·53-s + 245·57-s + 280·59-s + 518·61-s − 132·63-s + 141·67-s − 1.13e3·69-s + ⋯
 L(s)  = 1 + 1.34·3-s − 0.323·7-s + 0.814·9-s + 1.17·11-s − 0.597·13-s − 1.29·17-s + 0.422·19-s − 0.436·21-s − 1.46·23-s − 0.249·27-s − 1.02·29-s + 0.243·31-s + 1.58·33-s − 1.39·37-s − 0.804·39-s − 0.773·41-s + 0.326·43-s − 0.608·47-s − 0.895·49-s − 1.74·51-s + 0.212·53-s + 0.569·57-s + 0.617·59-s + 1.08·61-s − 0.263·63-s + 0.257·67-s − 1.97·69-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1600$$    =    $$2^{6} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$94.4030$$ Root analytic conductor: $$9.71612$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1600,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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$$
good3 $$1 - 7 T + p^{3} T^{2}$$
7 $$1 + 6 T + p^{3} T^{2}$$
11 $$1 - 43 T + p^{3} T^{2}$$
13 $$1 + 28 T + p^{3} T^{2}$$
17 $$1 + 91 T + p^{3} T^{2}$$
19 $$1 - 35 T + p^{3} T^{2}$$
23 $$1 + 162 T + p^{3} T^{2}$$
29 $$1 + 160 T + p^{3} T^{2}$$
31 $$1 - 42 T + p^{3} T^{2}$$
37 $$1 + 314 T + p^{3} T^{2}$$
41 $$1 + 203 T + p^{3} T^{2}$$
43 $$1 - 92 T + p^{3} T^{2}$$
47 $$1 + 196 T + p^{3} T^{2}$$
53 $$1 - 82 T + p^{3} T^{2}$$
59 $$1 - 280 T + p^{3} T^{2}$$
61 $$1 - 518 T + p^{3} T^{2}$$
67 $$1 - 141 T + p^{3} T^{2}$$
71 $$1 - 412 T + p^{3} T^{2}$$
73 $$1 - 763 T + p^{3} T^{2}$$
79 $$1 - 510 T + p^{3} T^{2}$$
83 $$1 - 777 T + p^{3} T^{2}$$
89 $$1 + 945 T + p^{3} T^{2}$$
97 $$1 + 1246 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$