# Properties

 Label 2-405-9.7-c3-0-25 Degree $2$ Conductor $405$ Sign $0.766 + 0.642i$ 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 + (2 − 3.46i)2-s + (−3.99 − 6.92i)4-s + (2.5 + 4.33i)5-s + (−3 + 5.19i)7-s + 20·10-s + (−16 + 27.7i)11-s + (19 + 32.9i)13-s + (12 + 20.7i)14-s + (31.9 − 55.4i)16-s + 26·17-s + 100·19-s + (20.0 − 34.6i)20-s + (63.9 + 110. i)22-s + (39 + 67.5i)23-s + (−12.5 + 21.6i)25-s + 152·26-s + ⋯
 L(s)  = 1 + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.223 + 0.387i)5-s + (−0.161 + 0.280i)7-s + 0.632·10-s + (−0.438 + 0.759i)11-s + (0.405 + 0.702i)13-s + (0.229 + 0.396i)14-s + (0.499 − 0.866i)16-s + 0.370·17-s + 1.20·19-s + (0.223 − 0.387i)20-s + (0.620 + 1.07i)22-s + (0.353 + 0.612i)23-s + (−0.100 + 0.173i)25-s + 1.14·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$405$$    =    $$3^{4} \cdot 5$$ Sign: $0.766 + 0.642i$ 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: $$(2,\ 405,\ (\ :3/2),\ 0.766 + 0.642i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$3.050824654$$ $$L(\frac12)$$ $$\approx$$ $$3.050824654$$ $$L(\frac{5}{2})$$ 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 + (-2 + 3.46i)T + (-4 - 6.92i)T^{2}$$
7 $$1 + (3 - 5.19i)T + (-171.5 - 297. i)T^{2}$$
11 $$1 + (16 - 27.7i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + (-19 - 32.9i)T + (-1.09e3 + 1.90e3i)T^{2}$$
17 $$1 - 26T + 4.91e3T^{2}$$
19 $$1 - 100T + 6.85e3T^{2}$$
23 $$1 + (-39 - 67.5i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (-25 + 43.3i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + (-54 - 93.5i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 - 266T + 5.06e4T^{2}$$
41 $$1 + (11 + 19.0i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (221 - 382. i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + (-257 + 445. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 - 2T + 1.48e5T^{2}$$
59 $$1 + (250 + 433. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-259 + 448. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (63 + 109. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 412T + 3.57e5T^{2}$$
73 $$1 + 878T + 3.89e5T^{2}$$
79 $$1 + (300 - 519. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (141 - 244. i)T + (-2.85e5 - 4.95e5i)T^{2}$$
89 $$1 + 150T + 7.04e5T^{2}$$
97 $$1 + (193 - 334. i)T + (-4.56e5 - 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$