# Properties

 Label 2-1800-1.1-c3-0-16 Degree $2$ Conductor $1800$ Sign $1$ Analytic cond. $106.203$ Root an. cond. $10.3055$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 4·11-s − 54·13-s + 114·17-s + 44·19-s + 96·23-s − 134·29-s − 272·31-s + 98·37-s + 6·41-s − 12·43-s − 200·47-s − 343·49-s + 654·53-s − 36·59-s − 442·61-s + 188·67-s + 632·71-s + 390·73-s + 688·79-s + 1.18e3·83-s + 694·89-s + 1.72e3·97-s − 1.18e3·101-s − 1.96e3·103-s + 796·107-s + 342·109-s + 114·113-s + ⋯
 L(s)  = 1 − 0.109·11-s − 1.15·13-s + 1.62·17-s + 0.531·19-s + 0.870·23-s − 0.858·29-s − 1.57·31-s + 0.435·37-s + 0.0228·41-s − 0.0425·43-s − 0.620·47-s − 49-s + 1.69·53-s − 0.0794·59-s − 0.927·61-s + 0.342·67-s + 1.05·71-s + 0.625·73-s + 0.979·79-s + 1.57·83-s + 0.826·89-s + 1.80·97-s − 1.16·101-s − 1.88·103-s + 0.719·107-s + 0.300·109-s + 0.0949·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1800$$    =    $$2^{3} \cdot 3^{2} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$106.203$$ Root analytic conductor: $$10.3055$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1800,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.953187726$$ $$L(\frac12)$$ $$\approx$$ $$1.953187726$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
good7 $$1 + p^{3} T^{2}$$
11 $$1 + 4 T + p^{3} T^{2}$$
13 $$1 + 54 T + p^{3} T^{2}$$
17 $$1 - 114 T + p^{3} T^{2}$$
19 $$1 - 44 T + p^{3} T^{2}$$
23 $$1 - 96 T + p^{3} T^{2}$$
29 $$1 + 134 T + p^{3} T^{2}$$
31 $$1 + 272 T + p^{3} T^{2}$$
37 $$1 - 98 T + p^{3} T^{2}$$
41 $$1 - 6 T + p^{3} T^{2}$$
43 $$1 + 12 T + p^{3} T^{2}$$
47 $$1 + 200 T + p^{3} T^{2}$$
53 $$1 - 654 T + p^{3} T^{2}$$
59 $$1 + 36 T + p^{3} T^{2}$$
61 $$1 + 442 T + p^{3} T^{2}$$
67 $$1 - 188 T + p^{3} T^{2}$$
71 $$1 - 632 T + p^{3} T^{2}$$
73 $$1 - 390 T + p^{3} T^{2}$$
79 $$1 - 688 T + p^{3} T^{2}$$
83 $$1 - 1188 T + p^{3} T^{2}$$
89 $$1 - 694 T + p^{3} T^{2}$$
97 $$1 - 1726 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$