# Properties

 Label 2-405-9.7-c3-0-16 Degree $2$ Conductor $405$ Sign $0.173 - 0.984i$ Analytic cond. $23.8957$ Root an. cond. $4.88833$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)2-s + (3.5 + 6.06i)4-s + (2.5 + 4.33i)5-s + (3 − 5.19i)7-s + 15·8-s + 5·10-s + (−23.5 + 40.7i)11-s + (2.5 + 4.33i)13-s + (−3 − 5.19i)14-s + (−20.5 + 35.5i)16-s + 131·17-s − 56·19-s + (−17.5 + 30.3i)20-s + (23.5 + 40.7i)22-s + (1.5 + 2.59i)23-s + ⋯
 L(s)  = 1 + (0.176 − 0.306i)2-s + (0.437 + 0.757i)4-s + (0.223 + 0.387i)5-s + (0.161 − 0.280i)7-s + 0.662·8-s + 0.158·10-s + (−0.644 + 1.11i)11-s + (0.0533 + 0.0923i)13-s + (−0.0572 − 0.0991i)14-s + (−0.320 + 0.554i)16-s + 1.86·17-s − 0.676·19-s + (−0.195 + 0.338i)20-s + (0.227 + 0.394i)22-s + (0.0135 + 0.0235i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$405$$    =    $$3^{4} \cdot 5$$ Sign: $0.173 - 0.984i$ Analytic conductor: $$23.8957$$ Root analytic conductor: $$4.88833$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{405} (136, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 405,\ (\ :3/2),\ 0.173 - 0.984i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.272161186$$ $$L(\frac12)$$ $$\approx$$ $$2.272161186$$ $$L(\frac{5}{2})$$ 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 + (-2.5 - 4.33i)T$$
good2 $$1 + (-0.5 + 0.866i)T + (-4 - 6.92i)T^{2}$$
7 $$1 + (-3 + 5.19i)T + (-171.5 - 297. i)T^{2}$$
11 $$1 + (23.5 - 40.7i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + (-2.5 - 4.33i)T + (-1.09e3 + 1.90e3i)T^{2}$$
17 $$1 - 131T + 4.91e3T^{2}$$
19 $$1 + 56T + 6.85e3T^{2}$$
23 $$1 + (-1.5 - 2.59i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (78.5 - 135. i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + (112.5 + 194. i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + 70T + 5.06e4T^{2}$$
41 $$1 + (-70 - 121. i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (198.5 - 343. i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + (173.5 - 300. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + 4T + 1.48e5T^{2}$$
59 $$1 + (-374 - 647. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-169 + 292. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (246 + 426. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 32T + 3.57e5T^{2}$$
73 $$1 - 970T + 3.89e5T^{2}$$
79 $$1 + (-628.5 + 1.08e3i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (51 - 88.3i)T + (-2.85e5 - 4.95e5i)T^{2}$$
89 $$1 - 1.48e3T + 7.04e5T^{2}$$
97 $$1 + (487 - 843. i)T + (-4.56e5 - 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$