# Properties

 Label 2-39e2-1.1-c3-0-119 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 $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.85·2-s + 6.86·4-s + 19.5·5-s + 6.26·7-s − 4.36·8-s + 75.3·10-s + 27.2·11-s + 24.1·14-s − 71.7·16-s + 30.8·17-s − 127.·19-s + 134.·20-s + 105.·22-s + 84.3·23-s + 256.·25-s + 43.0·28-s + 272.·29-s + 166.·31-s − 241.·32-s + 118.·34-s + 122.·35-s + 198.·37-s − 492.·38-s − 85.2·40-s − 160.·41-s + 158.·43-s + 187.·44-s + ⋯
 L(s)  = 1 + 1.36·2-s + 0.858·4-s + 1.74·5-s + 0.338·7-s − 0.192·8-s + 2.38·10-s + 0.746·11-s + 0.461·14-s − 1.12·16-s + 0.440·17-s − 1.54·19-s + 1.49·20-s + 1.01·22-s + 0.764·23-s + 2.05·25-s + 0.290·28-s + 1.74·29-s + 0.963·31-s − 1.33·32-s + 0.599·34-s + 0.590·35-s + 0.880·37-s − 2.10·38-s − 0.337·40-s − 0.610·41-s + 0.563·43-s + 0.641·44-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: $$0$$ Selberg data: $$(2,\ 1521,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$7.090585938$$ $$L(\frac12)$$ $$\approx$$ $$7.090585938$$ $$L(\frac{5}{2})$$ 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 - 3.85T + 8T^{2}$$
5 $$1 - 19.5T + 125T^{2}$$
7 $$1 - 6.26T + 343T^{2}$$
11 $$1 - 27.2T + 1.33e3T^{2}$$
17 $$1 - 30.8T + 4.91e3T^{2}$$
19 $$1 + 127.T + 6.85e3T^{2}$$
23 $$1 - 84.3T + 1.21e4T^{2}$$
29 $$1 - 272.T + 2.43e4T^{2}$$
31 $$1 - 166.T + 2.97e4T^{2}$$
37 $$1 - 198.T + 5.06e4T^{2}$$
41 $$1 + 160.T + 6.89e4T^{2}$$
43 $$1 - 158.T + 7.95e4T^{2}$$
47 $$1 - 305.T + 1.03e5T^{2}$$
53 $$1 - 356.T + 1.48e5T^{2}$$
59 $$1 - 470.T + 2.05e5T^{2}$$
61 $$1 + 171.T + 2.26e5T^{2}$$
67 $$1 + 1.02e3T + 3.00e5T^{2}$$
71 $$1 - 188.T + 3.57e5T^{2}$$
73 $$1 + 959.T + 3.89e5T^{2}$$
79 $$1 + 1.03e3T + 4.93e5T^{2}$$
83 $$1 - 105.T + 5.71e5T^{2}$$
89 $$1 - 649.T + 7.04e5T^{2}$$
97 $$1 + 707.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$