# Properties

 Label 2-1560-1.1-c3-0-59 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 − 35.1·7-s + 9·9-s + 13.7·11-s + 13·13-s + 15·15-s + 87.1·17-s + 11.6·19-s − 105.·21-s − 108.·23-s + 25·25-s + 27·27-s − 150.·29-s − 128.·31-s + 41.2·33-s − 175.·35-s − 11.7·37-s + 39·39-s + 366.·41-s + 301.·43-s + 45·45-s − 156.·47-s + 892.·49-s + 261.·51-s − 229.·53-s + 68.7·55-s + ⋯
 L(s)  = 1 + 0.577·3-s + 0.447·5-s − 1.89·7-s + 0.333·9-s + 0.376·11-s + 0.277·13-s + 0.258·15-s + 1.24·17-s + 0.141·19-s − 1.09·21-s − 0.981·23-s + 0.200·25-s + 0.192·27-s − 0.964·29-s − 0.742·31-s + 0.217·33-s − 0.848·35-s − 0.0523·37-s + 0.160·39-s + 1.39·41-s + 1.06·43-s + 0.149·45-s − 0.486·47-s + 2.60·49-s + 0.717·51-s − 0.593·53-s + 0.168·55-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 + 35.1T + 343T^{2}$$
11 $$1 - 13.7T + 1.33e3T^{2}$$
17 $$1 - 87.1T + 4.91e3T^{2}$$
19 $$1 - 11.6T + 6.85e3T^{2}$$
23 $$1 + 108.T + 1.21e4T^{2}$$
29 $$1 + 150.T + 2.43e4T^{2}$$
31 $$1 + 128.T + 2.97e4T^{2}$$
37 $$1 + 11.7T + 5.06e4T^{2}$$
41 $$1 - 366.T + 6.89e4T^{2}$$
43 $$1 - 301.T + 7.95e4T^{2}$$
47 $$1 + 156.T + 1.03e5T^{2}$$
53 $$1 + 229.T + 1.48e5T^{2}$$
59 $$1 + 377.T + 2.05e5T^{2}$$
61 $$1 + 406.T + 2.26e5T^{2}$$
67 $$1 + 352.T + 3.00e5T^{2}$$
71 $$1 + 842.T + 3.57e5T^{2}$$
73 $$1 - 50.4T + 3.89e5T^{2}$$
79 $$1 + 695.T + 4.93e5T^{2}$$
83 $$1 + 675.T + 5.71e5T^{2}$$
89 $$1 + 943.T + 7.04e5T^{2}$$
97 $$1 - 73.8T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$