# Properties

 Label 2-405-5.4-c3-0-65 Degree $2$ Conductor $405$ Sign $-0.522 - 0.852i$ 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 − 3.08i·2-s − 1.53·4-s + (−5.83 − 9.53i)5-s − 31.3i·7-s − 19.9i·8-s + (−29.4 + 18.0i)10-s − 19.1·11-s + 20.9i·13-s − 96.6·14-s − 73.9·16-s − 6.19i·17-s + 96.6·19-s + (8.94 + 14.6i)20-s + 59.2i·22-s − 162. i·23-s + ⋯
 L(s)  = 1 − 1.09i·2-s − 0.191·4-s + (−0.522 − 0.852i)5-s − 1.69i·7-s − 0.882i·8-s + (−0.930 + 0.569i)10-s − 0.526·11-s + 0.447i·13-s − 1.84·14-s − 1.15·16-s − 0.0883i·17-s + 1.16·19-s + (0.0999 + 0.163i)20-s + 0.574i·22-s − 1.47i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$405$$    =    $$3^{4} \cdot 5$$ Sign: $-0.522 - 0.852i$ Analytic conductor: $$23.8957$$ Root analytic conductor: $$4.88833$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{405} (244, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 405,\ (\ :3/2),\ -0.522 - 0.852i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.367666274$$ $$L(\frac12)$$ $$\approx$$ $$1.367666274$$ $$L(\frac{5}{2})$$ 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 + (5.83 + 9.53i)T$$
good2 $$1 + 3.08iT - 8T^{2}$$
7 $$1 + 31.3iT - 343T^{2}$$
11 $$1 + 19.1T + 1.33e3T^{2}$$
13 $$1 - 20.9iT - 2.19e3T^{2}$$
17 $$1 + 6.19iT - 4.91e3T^{2}$$
19 $$1 - 96.6T + 6.85e3T^{2}$$
23 $$1 + 162. iT - 1.21e4T^{2}$$
29 $$1 + 7.64T + 2.43e4T^{2}$$
31 $$1 - 225.T + 2.97e4T^{2}$$
37 $$1 - 155. iT - 5.06e4T^{2}$$
41 $$1 + 315.T + 6.89e4T^{2}$$
43 $$1 - 192. iT - 7.95e4T^{2}$$
47 $$1 - 318. iT - 1.03e5T^{2}$$
53 $$1 + 277. iT - 1.48e5T^{2}$$
59 $$1 - 429.T + 2.05e5T^{2}$$
61 $$1 + 89.9T + 2.26e5T^{2}$$
67 $$1 - 583. iT - 3.00e5T^{2}$$
71 $$1 - 132.T + 3.57e5T^{2}$$
73 $$1 + 259. iT - 3.89e5T^{2}$$
79 $$1 + 124.T + 4.93e5T^{2}$$
83 $$1 + 36.7iT - 5.71e5T^{2}$$
89 $$1 + 333.T + 7.04e5T^{2}$$
97 $$1 + 1.02e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$