# Properties

 Label 2-39e2-1.1-c3-0-137 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 + 5.36·2-s + 20.7·4-s − 2.69·5-s + 15.2·7-s + 68.5·8-s − 14.4·10-s + 66.8·11-s + 81.5·14-s + 201.·16-s − 4.16·17-s + 26.0·19-s − 56.0·20-s + 358.·22-s − 47.3·23-s − 117.·25-s + 315.·28-s − 257.·29-s + 206.·31-s + 532.·32-s − 22.3·34-s − 40.9·35-s + 175.·37-s + 139.·38-s − 184.·40-s − 156.·41-s + 51.9·43-s + 1.38e3·44-s + ⋯
 L(s)  = 1 + 1.89·2-s + 2.59·4-s − 0.241·5-s + 0.820·7-s + 3.03·8-s − 0.457·10-s + 1.83·11-s + 1.55·14-s + 3.14·16-s − 0.0594·17-s + 0.314·19-s − 0.626·20-s + 3.47·22-s − 0.429·23-s − 0.941·25-s + 2.13·28-s − 1.64·29-s + 1.19·31-s + 2.94·32-s − 0.112·34-s − 0.197·35-s + 0.780·37-s + 0.597·38-s − 0.730·40-s − 0.595·41-s + 0.184·43-s + 4.76·44-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

 $$L(2)$$ $$\approx$$ $$9.669151868$$ $$L(\frac12)$$ $$\approx$$ $$9.669151868$$ $$L(\frac{5}{2})$$ 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 - 5.36T + 8T^{2}$$
5 $$1 + 2.69T + 125T^{2}$$
7 $$1 - 15.2T + 343T^{2}$$
11 $$1 - 66.8T + 1.33e3T^{2}$$
17 $$1 + 4.16T + 4.91e3T^{2}$$
19 $$1 - 26.0T + 6.85e3T^{2}$$
23 $$1 + 47.3T + 1.21e4T^{2}$$
29 $$1 + 257.T + 2.43e4T^{2}$$
31 $$1 - 206.T + 2.97e4T^{2}$$
37 $$1 - 175.T + 5.06e4T^{2}$$
41 $$1 + 156.T + 6.89e4T^{2}$$
43 $$1 - 51.9T + 7.95e4T^{2}$$
47 $$1 - 354.T + 1.03e5T^{2}$$
53 $$1 - 10.4T + 1.48e5T^{2}$$
59 $$1 + 445.T + 2.05e5T^{2}$$
61 $$1 - 119.T + 2.26e5T^{2}$$
67 $$1 + 22.4T + 3.00e5T^{2}$$
71 $$1 + 285.T + 3.57e5T^{2}$$
73 $$1 - 740.T + 3.89e5T^{2}$$
79 $$1 + 547.T + 4.93e5T^{2}$$
83 $$1 + 603.T + 5.71e5T^{2}$$
89 $$1 + 215.T + 7.04e5T^{2}$$
97 $$1 - 1.44e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$