# Properties

 Label 2-2760-1.1-c1-0-23 Degree $2$ Conductor $2760$ Sign $1$ Analytic cond. $22.0387$ Root an. cond. $4.69454$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3-s + 5-s + 1.39·7-s + 9-s + 6.64·13-s + 15-s + 3.39·17-s + 1.39·21-s + 23-s + 25-s + 27-s + 0.601·29-s − 6.04·31-s + 1.39·35-s − 8.69·37-s + 6.64·39-s + 5.24·41-s − 7.44·43-s + 45-s + 2.79·47-s − 5.04·49-s + 3.39·51-s + 9.24·53-s − 3.24·59-s − 9.44·61-s + 1.39·63-s + 6.64·65-s + ⋯
 L(s)  = 1 + 0.577·3-s + 0.447·5-s + 0.528·7-s + 0.333·9-s + 1.84·13-s + 0.258·15-s + 0.824·17-s + 0.305·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.111·29-s − 1.08·31-s + 0.236·35-s − 1.42·37-s + 1.06·39-s + 0.819·41-s − 1.13·43-s + 0.149·45-s + 0.407·47-s − 0.720·49-s + 0.475·51-s + 1.27·53-s − 0.422·59-s − 1.20·61-s + 0.176·63-s + 0.824·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2760$$    =    $$2^{3} \cdot 3 \cdot 5 \cdot 23$$ Sign: $1$ Analytic conductor: $$22.0387$$ Root analytic conductor: $$4.69454$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2760,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$3.014017478$$ $$L(\frac12)$$ $$\approx$$ $$3.014017478$$ $$L(\frac{3}{2})$$ 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$$
23 $$1 - T$$
good7 $$1 - 1.39T + 7T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 - 6.64T + 13T^{2}$$
17 $$1 - 3.39T + 17T^{2}$$
19 $$1 + 19T^{2}$$
29 $$1 - 0.601T + 29T^{2}$$
31 $$1 + 6.04T + 31T^{2}$$
37 $$1 + 8.69T + 37T^{2}$$
41 $$1 - 5.24T + 41T^{2}$$
43 $$1 + 7.44T + 43T^{2}$$
47 $$1 - 2.79T + 47T^{2}$$
53 $$1 - 9.24T + 53T^{2}$$
59 $$1 + 3.24T + 59T^{2}$$
61 $$1 + 9.44T + 61T^{2}$$
67 $$1 - 10.6T + 67T^{2}$$
71 $$1 + 7.89T + 71T^{2}$$
73 $$1 - 4.79T + 73T^{2}$$
79 $$1 - 5.85T + 79T^{2}$$
83 $$1 - 11.8T + 83T^{2}$$
89 $$1 + 6.09T + 89T^{2}$$
97 $$1 - 9.44T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$