# Properties

 Label 2-39e2-1.1-c3-0-91 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 + 4.12·2-s + 9·4-s − 3.05·5-s + 6.68·7-s + 4.12·8-s − 12.5·10-s + 32.2·11-s + 27.5·14-s − 55·16-s + 28.8·17-s + 101.·19-s − 27.4·20-s + 132.·22-s + 118.·23-s − 115.·25-s + 60.1·28-s − 160.·29-s + 38.0·31-s − 259.·32-s + 118.·34-s − 20.3·35-s + 327.·37-s + 417.·38-s − 12.5·40-s + 56.0·41-s + 127.·43-s + 290.·44-s + ⋯
 L(s)  = 1 + 1.45·2-s + 1.12·4-s − 0.272·5-s + 0.360·7-s + 0.182·8-s − 0.397·10-s + 0.883·11-s + 0.526·14-s − 0.859·16-s + 0.411·17-s + 1.22·19-s − 0.306·20-s + 1.28·22-s + 1.07·23-s − 0.925·25-s + 0.405·28-s − 1.02·29-s + 0.220·31-s − 1.43·32-s + 0.600·34-s − 0.0984·35-s + 1.45·37-s + 1.78·38-s − 0.0497·40-s + 0.213·41-s + 0.453·43-s + 0.994·44-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

 $$L(2)$$ $$\approx$$ $$5.316941404$$ $$L(\frac12)$$ $$\approx$$ $$5.316941404$$ $$L(\frac{5}{2})$$ 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 - 4.12T + 8T^{2}$$
5 $$1 + 3.05T + 125T^{2}$$
7 $$1 - 6.68T + 343T^{2}$$
11 $$1 - 32.2T + 1.33e3T^{2}$$
17 $$1 - 28.8T + 4.91e3T^{2}$$
19 $$1 - 101.T + 6.85e3T^{2}$$
23 $$1 - 118.T + 1.21e4T^{2}$$
29 $$1 + 160.T + 2.43e4T^{2}$$
31 $$1 - 38.0T + 2.97e4T^{2}$$
37 $$1 - 327.T + 5.06e4T^{2}$$
41 $$1 - 56.0T + 6.89e4T^{2}$$
43 $$1 - 127.T + 7.95e4T^{2}$$
47 $$1 + 517.T + 1.03e5T^{2}$$
53 $$1 - 695.T + 1.48e5T^{2}$$
59 $$1 - 656.T + 2.05e5T^{2}$$
61 $$1 - 701.T + 2.26e5T^{2}$$
67 $$1 - 57.1T + 3.00e5T^{2}$$
71 $$1 - 309.T + 3.57e5T^{2}$$
73 $$1 - 389.T + 3.89e5T^{2}$$
79 $$1 - 901.T + 4.93e5T^{2}$$
83 $$1 - 687.T + 5.71e5T^{2}$$
89 $$1 - 1.07e3T + 7.04e5T^{2}$$
97 $$1 + 1.75e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$