# Properties

 Label 2-39e2-1.1-c3-0-184 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.45·2-s + 3.95·4-s + 9.27·5-s + 27.2·7-s − 13.9·8-s + 32.0·10-s − 38.3·11-s + 94.1·14-s − 79.9·16-s − 85.0·17-s − 114.·19-s + 36.7·20-s − 132.·22-s − 49.4·23-s − 38.8·25-s + 107.·28-s + 11.5·29-s − 220.·31-s − 164.·32-s − 294.·34-s + 252.·35-s − 24.7·37-s − 395.·38-s − 129.·40-s + 30.8·41-s − 409.·43-s − 151.·44-s + ⋯
 L(s)  = 1 + 1.22·2-s + 0.494·4-s + 0.829·5-s + 1.47·7-s − 0.617·8-s + 1.01·10-s − 1.05·11-s + 1.79·14-s − 1.24·16-s − 1.21·17-s − 1.38·19-s + 0.410·20-s − 1.28·22-s − 0.448·23-s − 0.311·25-s + 0.727·28-s + 0.0740·29-s − 1.27·31-s − 0.910·32-s − 1.48·34-s + 1.22·35-s − 0.109·37-s − 1.68·38-s − 0.512·40-s + 0.117·41-s − 1.45·43-s − 0.520·44-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: $$1$$ Selberg data: $$(2,\ 1521,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 - 3.45T + 8T^{2}$$
5 $$1 - 9.27T + 125T^{2}$$
7 $$1 - 27.2T + 343T^{2}$$
11 $$1 + 38.3T + 1.33e3T^{2}$$
17 $$1 + 85.0T + 4.91e3T^{2}$$
19 $$1 + 114.T + 6.85e3T^{2}$$
23 $$1 + 49.4T + 1.21e4T^{2}$$
29 $$1 - 11.5T + 2.43e4T^{2}$$
31 $$1 + 220.T + 2.97e4T^{2}$$
37 $$1 + 24.7T + 5.06e4T^{2}$$
41 $$1 - 30.8T + 6.89e4T^{2}$$
43 $$1 + 409.T + 7.95e4T^{2}$$
47 $$1 - 434.T + 1.03e5T^{2}$$
53 $$1 + 716.T + 1.48e5T^{2}$$
59 $$1 - 618.T + 2.05e5T^{2}$$
61 $$1 - 213.T + 2.26e5T^{2}$$
67 $$1 + 578.T + 3.00e5T^{2}$$
71 $$1 - 238.T + 3.57e5T^{2}$$
73 $$1 - 748.T + 3.89e5T^{2}$$
79 $$1 - 883.T + 4.93e5T^{2}$$
83 $$1 - 1.40e3T + 5.71e5T^{2}$$
89 $$1 + 1.30e3T + 7.04e5T^{2}$$
97 $$1 - 258.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$