# Properties

 Label 2-39e2-1.1-c3-0-84 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 − 5.52·2-s + 22.4·4-s + 6.08·5-s + 20.2·7-s − 80.0·8-s − 33.5·10-s + 48.8·11-s − 111.·14-s + 262.·16-s + 37.7·17-s + 120.·19-s + 136.·20-s − 269.·22-s − 74.8·23-s − 88.0·25-s + 456.·28-s + 112.·29-s + 113.·31-s − 807.·32-s − 208.·34-s + 123.·35-s − 85.7·37-s − 667.·38-s − 486.·40-s + 133.·41-s − 319.·43-s + 1.09e3·44-s + ⋯
 L(s)  = 1 − 1.95·2-s + 2.81·4-s + 0.543·5-s + 1.09·7-s − 3.53·8-s − 1.06·10-s + 1.33·11-s − 2.13·14-s + 4.09·16-s + 0.538·17-s + 1.45·19-s + 1.52·20-s − 2.61·22-s − 0.678·23-s − 0.704·25-s + 3.07·28-s + 0.721·29-s + 0.655·31-s − 4.46·32-s − 1.05·34-s + 0.595·35-s − 0.381·37-s − 2.84·38-s − 1.92·40-s + 0.510·41-s − 1.13·43-s + 3.76·44-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

 $$L(2)$$ $$\approx$$ $$1.462243442$$ $$L(\frac12)$$ $$\approx$$ $$1.462243442$$ $$L(\frac{5}{2})$$ 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 + 5.52T + 8T^{2}$$
5 $$1 - 6.08T + 125T^{2}$$
7 $$1 - 20.2T + 343T^{2}$$
11 $$1 - 48.8T + 1.33e3T^{2}$$
17 $$1 - 37.7T + 4.91e3T^{2}$$
19 $$1 - 120.T + 6.85e3T^{2}$$
23 $$1 + 74.8T + 1.21e4T^{2}$$
29 $$1 - 112.T + 2.43e4T^{2}$$
31 $$1 - 113.T + 2.97e4T^{2}$$
37 $$1 + 85.7T + 5.06e4T^{2}$$
41 $$1 - 133.T + 6.89e4T^{2}$$
43 $$1 + 319.T + 7.95e4T^{2}$$
47 $$1 - 401.T + 1.03e5T^{2}$$
53 $$1 - 384.T + 1.48e5T^{2}$$
59 $$1 - 121.T + 2.05e5T^{2}$$
61 $$1 - 220.T + 2.26e5T^{2}$$
67 $$1 - 975.T + 3.00e5T^{2}$$
71 $$1 + 106.T + 3.57e5T^{2}$$
73 $$1 - 43.2T + 3.89e5T^{2}$$
79 $$1 - 539.T + 4.93e5T^{2}$$
83 $$1 + 811.T + 5.71e5T^{2}$$
89 $$1 - 1.13e3T + 7.04e5T^{2}$$
97 $$1 + 229.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$