# Properties

 Label 2-405-1.1-c3-0-4 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 + 0.225·2-s − 7.94·4-s + 5·5-s − 31.1·7-s − 3.59·8-s + 1.12·10-s − 18.1·11-s − 50.1·13-s − 7.02·14-s + 62.7·16-s + 131.·17-s + 23.2·19-s − 39.7·20-s − 4.08·22-s + 32.9·23-s + 25·25-s − 11.2·26-s + 247.·28-s − 125.·29-s + 125.·31-s + 42.8·32-s + 29.6·34-s − 155.·35-s + 99.9·37-s + 5.23·38-s − 17.9·40-s − 245.·41-s + ⋯
 L(s)  = 1 + 0.0796·2-s − 0.993·4-s + 0.447·5-s − 1.68·7-s − 0.158·8-s + 0.0356·10-s − 0.496·11-s − 1.07·13-s − 0.134·14-s + 0.981·16-s + 1.87·17-s + 0.280·19-s − 0.444·20-s − 0.0395·22-s + 0.299·23-s + 0.200·25-s − 0.0852·26-s + 1.67·28-s − 0.805·29-s + 0.724·31-s + 0.236·32-s + 0.149·34-s − 0.753·35-s + 0.444·37-s + 0.0223·38-s − 0.0710·40-s − 0.934·41-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

 $$L(2)$$ $$\approx$$ $$1.058538194$$ $$L(\frac12)$$ $$\approx$$ $$1.058538194$$ $$L(\frac{5}{2})$$ 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 - 0.225T + 8T^{2}$$
7 $$1 + 31.1T + 343T^{2}$$
11 $$1 + 18.1T + 1.33e3T^{2}$$
13 $$1 + 50.1T + 2.19e3T^{2}$$
17 $$1 - 131.T + 4.91e3T^{2}$$
19 $$1 - 23.2T + 6.85e3T^{2}$$
23 $$1 - 32.9T + 1.21e4T^{2}$$
29 $$1 + 125.T + 2.43e4T^{2}$$
31 $$1 - 125.T + 2.97e4T^{2}$$
37 $$1 - 99.9T + 5.06e4T^{2}$$
41 $$1 + 245.T + 6.89e4T^{2}$$
43 $$1 - 139.T + 7.95e4T^{2}$$
47 $$1 - 472.T + 1.03e5T^{2}$$
53 $$1 + 421.T + 1.48e5T^{2}$$
59 $$1 - 742.T + 2.05e5T^{2}$$
61 $$1 - 8.97T + 2.26e5T^{2}$$
67 $$1 - 588.T + 3.00e5T^{2}$$
71 $$1 - 48.5T + 3.57e5T^{2}$$
73 $$1 - 409.T + 3.89e5T^{2}$$
79 $$1 - 530.T + 4.93e5T^{2}$$
83 $$1 - 294.T + 5.71e5T^{2}$$
89 $$1 + 852.T + 7.04e5T^{2}$$
97 $$1 + 388.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$