# Properties

 Label 2-4620-1.1-c1-0-39 Degree $2$ Conductor $4620$ Sign $-1$ Analytic cond. $36.8908$ Root an. cond. $6.07378$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 3-s + 5-s + 7-s + 9-s − 11-s − 4·13-s + 15-s + 3·17-s − 7·19-s + 21-s − 9·23-s + 25-s + 27-s − 3·29-s + 2·31-s − 33-s + 35-s − 4·37-s − 4·39-s − 6·41-s − 43-s + 45-s − 6·47-s + 49-s + 3·51-s + 3·53-s − 55-s + ⋯
 L(s)  = 1 + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.258·15-s + 0.727·17-s − 1.60·19-s + 0.218·21-s − 1.87·23-s + 1/5·25-s + 0.192·27-s − 0.557·29-s + 0.359·31-s − 0.174·33-s + 0.169·35-s − 0.657·37-s − 0.640·39-s − 0.937·41-s − 0.152·43-s + 0.149·45-s − 0.875·47-s + 1/7·49-s + 0.420·51-s + 0.412·53-s − 0.134·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4620$$    =    $$2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11$$ Sign: $-1$ Analytic conductor: $$36.8908$$ Root analytic conductor: $$6.07378$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 4620,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - T$$
5 $$1 - T$$
7 $$1 - T$$
11 $$1 + T$$
good13 $$1 + 4 T + p T^{2}$$
17 $$1 - 3 T + p T^{2}$$
19 $$1 + 7 T + p T^{2}$$
23 $$1 + 9 T + p T^{2}$$
29 $$1 + 3 T + p T^{2}$$
31 $$1 - 2 T + p T^{2}$$
37 $$1 + 4 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 + T + p T^{2}$$
47 $$1 + 6 T + p T^{2}$$
53 $$1 - 3 T + p T^{2}$$
59 $$1 + 9 T + p T^{2}$$
61 $$1 + T + p T^{2}$$
67 $$1 + 10 T + p T^{2}$$
71 $$1 - 6 T + p T^{2}$$
73 $$1 - 8 T + p T^{2}$$
79 $$1 + 4 T + p T^{2}$$
83 $$1 - 3 T + p T^{2}$$
89 $$1 + 15 T + p T^{2}$$
97 $$1 + T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$