# Properties

 Label 2-39e2-1.1-c3-0-104 Degree $2$ Conductor $1521$ Sign $-1$ Analytic cond. $89.7419$ Root an. cond. $9.47322$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.32·2-s − 6.25·4-s − 12.9·5-s − 5.54·7-s − 18.8·8-s − 17.1·10-s + 0.841·11-s − 7.33·14-s + 25.1·16-s + 10.1·17-s + 73.0·19-s + 81.0·20-s + 1.11·22-s + 139.·23-s + 42.9·25-s + 34.6·28-s + 250.·29-s + 161.·31-s + 183.·32-s + 13.4·34-s + 71.8·35-s − 195.·37-s + 96.6·38-s + 244.·40-s − 183.·41-s + 115.·43-s − 5.26·44-s + ⋯
 L(s)  = 1 + 0.467·2-s − 0.781·4-s − 1.15·5-s − 0.299·7-s − 0.832·8-s − 0.541·10-s + 0.0230·11-s − 0.139·14-s + 0.392·16-s + 0.145·17-s + 0.882·19-s + 0.905·20-s + 0.0107·22-s + 1.26·23-s + 0.343·25-s + 0.233·28-s + 1.60·29-s + 0.935·31-s + 1.01·32-s + 0.0679·34-s + 0.347·35-s − 0.870·37-s + 0.412·38-s + 0.965·40-s − 0.699·41-s + 0.408·43-s − 0.0180·44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1521$$    =    $$3^{2} \cdot 13^{2}$$ Sign: $-1$ Analytic conductor: $$89.7419$$ Root analytic conductor: $$9.47322$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1521,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad3 $$1$$
13 $$1$$
good2 $$1 - 1.32T + 8T^{2}$$
5 $$1 + 12.9T + 125T^{2}$$
7 $$1 + 5.54T + 343T^{2}$$
11 $$1 - 0.841T + 1.33e3T^{2}$$
17 $$1 - 10.1T + 4.91e3T^{2}$$
19 $$1 - 73.0T + 6.85e3T^{2}$$
23 $$1 - 139.T + 1.21e4T^{2}$$
29 $$1 - 250.T + 2.43e4T^{2}$$
31 $$1 - 161.T + 2.97e4T^{2}$$
37 $$1 + 195.T + 5.06e4T^{2}$$
41 $$1 + 183.T + 6.89e4T^{2}$$
43 $$1 - 115.T + 7.95e4T^{2}$$
47 $$1 + 551.T + 1.03e5T^{2}$$
53 $$1 + 324.T + 1.48e5T^{2}$$
59 $$1 - 521.T + 2.05e5T^{2}$$
61 $$1 + 444.T + 2.26e5T^{2}$$
67 $$1 + 55.0T + 3.00e5T^{2}$$
71 $$1 + 279.T + 3.57e5T^{2}$$
73 $$1 + 908.T + 3.89e5T^{2}$$
79 $$1 - 941.T + 4.93e5T^{2}$$
83 $$1 - 946.T + 5.71e5T^{2}$$
89 $$1 - 32.9T + 7.04e5T^{2}$$
97 $$1 - 68.2T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$