# Properties

 Label 2-42e2-1.1-c3-0-38 Degree $2$ Conductor $1764$ Sign $-1$ Analytic cond. $104.079$ Root an. cond. $10.2019$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 19·13-s + 107·19-s − 125·25-s − 289·31-s + 323·37-s + 71·43-s + 182·61-s − 127·67-s − 271·73-s − 1.38e3·79-s − 1.33e3·97-s − 1.80e3·103-s + 2.21e3·109-s + ⋯
 L(s)  = 1 − 0.405·13-s + 1.29·19-s − 25-s − 1.67·31-s + 1.43·37-s + 0.251·43-s + 0.382·61-s − 0.231·67-s − 0.434·73-s − 1.97·79-s − 1.39·97-s − 1.72·103-s + 1.94·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$104.079$$ Root analytic conductor: $$10.2019$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: $\chi_{1764} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1764,\ (\ :3/2),\ -1)$$

## 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$$
3 $$1$$
7 $$1$$
good5 $$1 + p^{3} T^{2}$$
11 $$1 + p^{3} T^{2}$$
13 $$1 + 19 T + p^{3} T^{2}$$
17 $$1 + p^{3} T^{2}$$
19 $$1 - 107 T + p^{3} T^{2}$$
23 $$1 + p^{3} T^{2}$$
29 $$1 + p^{3} T^{2}$$
31 $$1 + 289 T + p^{3} T^{2}$$
37 $$1 - 323 T + p^{3} T^{2}$$
41 $$1 + p^{3} T^{2}$$
43 $$1 - 71 T + p^{3} T^{2}$$
47 $$1 + p^{3} T^{2}$$
53 $$1 + p^{3} T^{2}$$
59 $$1 + p^{3} T^{2}$$
61 $$1 - 182 T + p^{3} T^{2}$$
67 $$1 + 127 T + p^{3} T^{2}$$
71 $$1 + p^{3} T^{2}$$
73 $$1 + 271 T + p^{3} T^{2}$$
79 $$1 + 1387 T + p^{3} T^{2}$$
83 $$1 + p^{3} T^{2}$$
89 $$1 + p^{3} T^{2}$$
97 $$1 + 1330 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$