# Properties

 Label 2-2760-1.1-c1-0-35 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3-s − 5-s + 4.46·7-s + 9-s + 3.39·11-s − 4.70·13-s + 15-s − 6.54·17-s − 6.22·19-s − 4.46·21-s + 23-s + 25-s − 27-s + 0.448·29-s − 5.15·31-s − 3.39·33-s − 4.46·35-s − 5.98·37-s + 4.70·39-s + 3.07·41-s − 4.47·43-s − 45-s + 6.22·47-s + 12.9·49-s + 6.54·51-s − 2.59·53-s − 3.39·55-s + ⋯
 L(s)  = 1 − 0.577·3-s − 0.447·5-s + 1.68·7-s + 0.333·9-s + 1.02·11-s − 1.30·13-s + 0.258·15-s − 1.58·17-s − 1.42·19-s − 0.974·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.0832·29-s − 0.925·31-s − 0.590·33-s − 0.754·35-s − 0.984·37-s + 0.753·39-s + 0.479·41-s − 0.683·43-s − 0.149·45-s + 0.908·47-s + 1.84·49-s + 0.916·51-s − 0.355·53-s − 0.457·55-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: $$1$$ Selberg data: $$(2,\ 2760,\ (\ :1/2),\ -1)$$

## 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$$
23 $$1 - T$$
good7 $$1 - 4.46T + 7T^{2}$$
11 $$1 - 3.39T + 11T^{2}$$
13 $$1 + 4.70T + 13T^{2}$$
17 $$1 + 6.54T + 17T^{2}$$
19 $$1 + 6.22T + 19T^{2}$$
29 $$1 - 0.448T + 29T^{2}$$
31 $$1 + 5.15T + 31T^{2}$$
37 $$1 + 5.98T + 37T^{2}$$
41 $$1 - 3.07T + 41T^{2}$$
43 $$1 + 4.47T + 43T^{2}$$
47 $$1 - 6.22T + 47T^{2}$$
53 $$1 + 2.59T + 53T^{2}$$
59 $$1 + 1.55T + 59T^{2}$$
61 $$1 + 4.91T + 61T^{2}$$
67 $$1 - 6.08T + 67T^{2}$$
71 $$1 - 1.07T + 71T^{2}$$
73 $$1 + 8.70T + 73T^{2}$$
79 $$1 + 13.1T + 79T^{2}$$
83 $$1 + 13.0T + 83T^{2}$$
89 $$1 - 11.6T + 89T^{2}$$
97 $$1 - 4.30T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$