# Properties

 Label 2-40e2-1.1-c3-0-36 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 $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 9·3-s − 26·7-s + 54·9-s − 59·11-s − 28·13-s + 5·17-s + 109·19-s − 234·21-s + 194·23-s + 243·27-s + 32·29-s − 10·31-s − 531·33-s + 198·37-s − 252·39-s + 117·41-s + 388·43-s + 68·47-s + 333·49-s + 45·51-s + 18·53-s + 981·57-s + 392·59-s + 710·61-s − 1.40e3·63-s − 253·67-s + 1.74e3·69-s + ⋯
 L(s)  = 1 + 1.73·3-s − 1.40·7-s + 2·9-s − 1.61·11-s − 0.597·13-s + 0.0713·17-s + 1.31·19-s − 2.43·21-s + 1.75·23-s + 1.73·27-s + 0.204·29-s − 0.0579·31-s − 2.80·33-s + 0.879·37-s − 1.03·39-s + 0.445·41-s + 1.37·43-s + 0.211·47-s + 0.970·49-s + 0.123·51-s + 0.0466·53-s + 2.27·57-s + 0.864·59-s + 1.49·61-s − 2.80·63-s − 0.461·67-s + 3.04·69-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: $$0$$ Selberg data: $$(2,\ 1600,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$3.333950470$$ $$L(\frac12)$$ $$\approx$$ $$3.333950470$$ $$L(\frac{5}{2})$$ 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 - p^{2} T + p^{3} T^{2}$$
7 $$1 + 26 T + p^{3} T^{2}$$
11 $$1 + 59 T + p^{3} T^{2}$$
13 $$1 + 28 T + p^{3} T^{2}$$
17 $$1 - 5 T + p^{3} T^{2}$$
19 $$1 - 109 T + p^{3} T^{2}$$
23 $$1 - 194 T + p^{3} T^{2}$$
29 $$1 - 32 T + p^{3} T^{2}$$
31 $$1 + 10 T + p^{3} T^{2}$$
37 $$1 - 198 T + p^{3} T^{2}$$
41 $$1 - 117 T + p^{3} T^{2}$$
43 $$1 - 388 T + p^{3} T^{2}$$
47 $$1 - 68 T + p^{3} T^{2}$$
53 $$1 - 18 T + p^{3} T^{2}$$
59 $$1 - 392 T + p^{3} T^{2}$$
61 $$1 - 710 T + p^{3} T^{2}$$
67 $$1 + 253 T + p^{3} T^{2}$$
71 $$1 - 612 T + p^{3} T^{2}$$
73 $$1 + 549 T + p^{3} T^{2}$$
79 $$1 + 414 T + p^{3} T^{2}$$
83 $$1 + 121 T + p^{3} T^{2}$$
89 $$1 + 81 T + p^{3} T^{2}$$
97 $$1 + 1502 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$