# Properties

 Label 2-4140-1.1-c1-0-21 Degree $2$ Conductor $4140$ Sign $1$ Analytic cond. $33.0580$ Root an. cond. $5.74961$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5-s + 5.21·7-s + 2.67·11-s + 0.186·13-s − 1.99·17-s − 0.675·19-s + 23-s + 25-s + 5.02·29-s + 7.52·31-s + 5.21·35-s + 4.34·37-s − 6.60·41-s − 2.84·43-s − 13.5·47-s + 20.1·49-s + 12.9·53-s + 2.67·55-s − 6.74·59-s + 8.65·61-s + 0.186·65-s − 12.1·67-s + 0.181·71-s + 3.81·73-s + 13.9·77-s − 10.4·79-s + 16.8·83-s + ⋯
 L(s)  = 1 + 0.447·5-s + 1.96·7-s + 0.806·11-s + 0.0516·13-s − 0.482·17-s − 0.154·19-s + 0.208·23-s + 0.200·25-s + 0.933·29-s + 1.35·31-s + 0.880·35-s + 0.715·37-s − 1.03·41-s − 0.433·43-s − 1.98·47-s + 2.87·49-s + 1.77·53-s + 0.360·55-s − 0.878·59-s + 1.10·61-s + 0.0230·65-s − 1.48·67-s + 0.0215·71-s + 0.446·73-s + 1.58·77-s − 1.18·79-s + 1.84·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4140$$    =    $$2^{2} \cdot 3^{2} \cdot 5 \cdot 23$$ Sign: $1$ Analytic conductor: $$33.0580$$ Root analytic conductor: $$5.74961$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 4140,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.001854418$$ $$L(\frac12)$$ $$\approx$$ $$3.001854418$$ $$L(\frac{3}{2})$$ 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$$
5 $$1 - T$$
23 $$1 - T$$
good7 $$1 - 5.21T + 7T^{2}$$
11 $$1 - 2.67T + 11T^{2}$$
13 $$1 - 0.186T + 13T^{2}$$
17 $$1 + 1.99T + 17T^{2}$$
19 $$1 + 0.675T + 19T^{2}$$
29 $$1 - 5.02T + 29T^{2}$$
31 $$1 - 7.52T + 31T^{2}$$
37 $$1 - 4.34T + 37T^{2}$$
41 $$1 + 6.60T + 41T^{2}$$
43 $$1 + 2.84T + 43T^{2}$$
47 $$1 + 13.5T + 47T^{2}$$
53 $$1 - 12.9T + 53T^{2}$$
59 $$1 + 6.74T + 59T^{2}$$
61 $$1 - 8.65T + 61T^{2}$$
67 $$1 + 12.1T + 67T^{2}$$
71 $$1 - 0.181T + 71T^{2}$$
73 $$1 - 3.81T + 73T^{2}$$
79 $$1 + 10.4T + 79T^{2}$$
83 $$1 - 16.8T + 83T^{2}$$
89 $$1 + 12.4T + 89T^{2}$$
97 $$1 + 4.92T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$