# Properties

 Label 2-39e2-1.1-c3-0-0 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 − 5.39·2-s + 21.1·4-s − 13.0·5-s + 6.42·7-s − 70.7·8-s + 70.3·10-s − 26.2·11-s − 34.6·14-s + 212.·16-s − 123.·17-s − 109.·19-s − 275.·20-s + 141.·22-s − 63.4·23-s + 45.0·25-s + 135.·28-s + 225.·29-s − 200.·31-s − 582.·32-s + 668.·34-s − 83.7·35-s − 252.·37-s + 591.·38-s + 922.·40-s − 227.·41-s − 384.·43-s − 553.·44-s + ⋯
 L(s)  = 1 − 1.90·2-s + 2.63·4-s − 1.16·5-s + 0.346·7-s − 3.12·8-s + 2.22·10-s − 0.718·11-s − 0.661·14-s + 3.32·16-s − 1.76·17-s − 1.32·19-s − 3.07·20-s + 1.37·22-s − 0.574·23-s + 0.360·25-s + 0.915·28-s + 1.44·29-s − 1.16·31-s − 3.21·32-s + 3.37·34-s − 0.404·35-s − 1.12·37-s + 2.52·38-s + 3.64·40-s − 0.866·41-s − 1.36·43-s − 1.89·44-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

 $$L(2)$$ $$\approx$$ $$0.01263014489$$ $$L(\frac12)$$ $$\approx$$ $$0.01263014489$$ $$L(\frac{5}{2})$$ 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 + 5.39T + 8T^{2}$$
5 $$1 + 13.0T + 125T^{2}$$
7 $$1 - 6.42T + 343T^{2}$$
11 $$1 + 26.2T + 1.33e3T^{2}$$
17 $$1 + 123.T + 4.91e3T^{2}$$
19 $$1 + 109.T + 6.85e3T^{2}$$
23 $$1 + 63.4T + 1.21e4T^{2}$$
29 $$1 - 225.T + 2.43e4T^{2}$$
31 $$1 + 200.T + 2.97e4T^{2}$$
37 $$1 + 252.T + 5.06e4T^{2}$$
41 $$1 + 227.T + 6.89e4T^{2}$$
43 $$1 + 384.T + 7.95e4T^{2}$$
47 $$1 + 34.6T + 1.03e5T^{2}$$
53 $$1 + 61.0T + 1.48e5T^{2}$$
59 $$1 - 80.5T + 2.05e5T^{2}$$
61 $$1 + 26.1T + 2.26e5T^{2}$$
67 $$1 - 931.T + 3.00e5T^{2}$$
71 $$1 + 427.T + 3.57e5T^{2}$$
73 $$1 + 108.T + 3.89e5T^{2}$$
79 $$1 - 384.T + 4.93e5T^{2}$$
83 $$1 - 85.9T + 5.71e5T^{2}$$
89 $$1 - 495.T + 7.04e5T^{2}$$
97 $$1 + 190.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$