# Properties

 Label 2-1560-1.1-c3-0-70 Degree $2$ Conductor $1560$ Sign $-1$ Analytic cond. $92.0429$ Root an. cond. $9.59390$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3·3-s + 5·5-s + 11.0·7-s + 9·9-s − 35.6·11-s + 13·13-s + 15·15-s + 9.34·17-s − 77.5·19-s + 33.1·21-s − 204.·23-s + 25·25-s + 27·27-s + 176.·29-s − 216.·31-s − 106.·33-s + 55.3·35-s − 190.·37-s + 39·39-s − 315.·41-s + 33.6·43-s + 45·45-s − 203.·47-s − 220.·49-s + 28.0·51-s + 487.·53-s − 178.·55-s + ⋯
 L(s)  = 1 + 0.577·3-s + 0.447·5-s + 0.597·7-s + 0.333·9-s − 0.977·11-s + 0.277·13-s + 0.258·15-s + 0.133·17-s − 0.936·19-s + 0.344·21-s − 1.85·23-s + 0.200·25-s + 0.192·27-s + 1.12·29-s − 1.25·31-s − 0.564·33-s + 0.267·35-s − 0.847·37-s + 0.160·39-s − 1.20·41-s + 0.119·43-s + 0.149·45-s − 0.630·47-s − 0.643·49-s + 0.0770·51-s + 1.26·53-s − 0.437·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1560$$    =    $$2^{3} \cdot 3 \cdot 5 \cdot 13$$ Sign: $-1$ Analytic conductor: $$92.0429$$ Root analytic conductor: $$9.59390$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1560,\ (\ :3/2),\ -1)$$

## 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 - 3T$$
5 $$1 - 5T$$
13 $$1 - 13T$$
good7 $$1 - 11.0T + 343T^{2}$$
11 $$1 + 35.6T + 1.33e3T^{2}$$
17 $$1 - 9.34T + 4.91e3T^{2}$$
19 $$1 + 77.5T + 6.85e3T^{2}$$
23 $$1 + 204.T + 1.21e4T^{2}$$
29 $$1 - 176.T + 2.43e4T^{2}$$
31 $$1 + 216.T + 2.97e4T^{2}$$
37 $$1 + 190.T + 5.06e4T^{2}$$
41 $$1 + 315.T + 6.89e4T^{2}$$
43 $$1 - 33.6T + 7.95e4T^{2}$$
47 $$1 + 203.T + 1.03e5T^{2}$$
53 $$1 - 487.T + 1.48e5T^{2}$$
59 $$1 - 176.T + 2.05e5T^{2}$$
61 $$1 + 234.T + 2.26e5T^{2}$$
67 $$1 + 608.T + 3.00e5T^{2}$$
71 $$1 + 454.T + 3.57e5T^{2}$$
73 $$1 + 427.T + 3.89e5T^{2}$$
79 $$1 - 1.17e3T + 4.93e5T^{2}$$
83 $$1 - 821.T + 5.71e5T^{2}$$
89 $$1 + 549.T + 7.04e5T^{2}$$
97 $$1 - 213.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$