# Properties

 Label 2-405-1.1-c3-0-8 Degree $2$ Conductor $405$ Sign $1$ Analytic cond. $23.8957$ Root an. cond. $4.88833$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.04·2-s + 1.25·4-s + 5·5-s − 13.7·7-s + 20.5·8-s − 15.2·10-s − 31.8·11-s + 58.2·13-s + 41.7·14-s − 72.4·16-s − 109.·17-s + 129.·19-s + 6.26·20-s + 96.7·22-s − 79.6·23-s + 25·25-s − 177.·26-s − 17.1·28-s − 9.03·29-s − 33.3·31-s + 56.1·32-s + 331.·34-s − 68.5·35-s − 22.1·37-s − 394.·38-s + 102.·40-s − 121.·41-s + ⋯
 L(s)  = 1 − 1.07·2-s + 0.156·4-s + 0.447·5-s − 0.740·7-s + 0.907·8-s − 0.480·10-s − 0.871·11-s + 1.24·13-s + 0.796·14-s − 1.13·16-s − 1.55·17-s + 1.56·19-s + 0.0699·20-s + 0.937·22-s − 0.722·23-s + 0.200·25-s − 1.33·26-s − 0.115·28-s − 0.0578·29-s − 0.193·31-s + 0.310·32-s + 1.67·34-s − 0.331·35-s − 0.0984·37-s − 1.68·38-s + 0.405·40-s − 0.463·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$405$$    =    $$3^{4} \cdot 5$$ Sign: $1$ Analytic conductor: $$23.8957$$ Root analytic conductor: $$4.88833$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{405} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 405,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.8236789412$$ $$L(\frac12)$$ $$\approx$$ $$0.8236789412$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1 - 5T$$
good2 $$1 + 3.04T + 8T^{2}$$
7 $$1 + 13.7T + 343T^{2}$$
11 $$1 + 31.8T + 1.33e3T^{2}$$
13 $$1 - 58.2T + 2.19e3T^{2}$$
17 $$1 + 109.T + 4.91e3T^{2}$$
19 $$1 - 129.T + 6.85e3T^{2}$$
23 $$1 + 79.6T + 1.21e4T^{2}$$
29 $$1 + 9.03T + 2.43e4T^{2}$$
31 $$1 + 33.3T + 2.97e4T^{2}$$
37 $$1 + 22.1T + 5.06e4T^{2}$$
41 $$1 + 121.T + 6.89e4T^{2}$$
43 $$1 - 10.1T + 7.95e4T^{2}$$
47 $$1 - 441.T + 1.03e5T^{2}$$
53 $$1 - 593.T + 1.48e5T^{2}$$
59 $$1 + 442.T + 2.05e5T^{2}$$
61 $$1 - 144.T + 2.26e5T^{2}$$
67 $$1 - 862.T + 3.00e5T^{2}$$
71 $$1 - 818.T + 3.57e5T^{2}$$
73 $$1 - 495.T + 3.89e5T^{2}$$
79 $$1 - 1.17e3T + 4.93e5T^{2}$$
83 $$1 - 424.T + 5.71e5T^{2}$$
89 $$1 + 1.03e3T + 7.04e5T^{2}$$
97 $$1 - 1.59e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$