Properties

 Label 2-405-9.4-c3-0-34 Degree $2$ Conductor $405$ Sign $0.939 + 0.342i$ 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 + (1 + 1.73i)2-s + (2.00 − 3.46i)4-s + (−2.5 + 4.33i)5-s + 24·8-s − 10·10-s + (−5 − 8.66i)11-s + (40 − 69.2i)13-s + (8.00 + 13.8i)16-s + 7·17-s − 113·19-s + (10 + 17.3i)20-s + (10 − 17.3i)22-s + (40.5 − 70.1i)23-s + (−12.5 − 21.6i)25-s + 160·26-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 1.06·8-s − 0.316·10-s + (−0.137 − 0.237i)11-s + (0.853 − 1.47i)13-s + (0.125 + 0.216i)16-s + 0.0998·17-s − 1.36·19-s + (0.111 + 0.193i)20-s + (0.0969 − 0.167i)22-s + (0.367 − 0.635i)23-s + (−0.100 − 0.173i)25-s + 1.20·26-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$405$$    =    $$3^{4} \cdot 5$$ Sign: $0.939 + 0.342i$ Analytic conductor: $$23.8957$$ Root analytic conductor: $$4.88833$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{405} (271, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 405,\ (\ :3/2),\ 0.939 + 0.342i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$2.475642903$$ $$L(\frac12)$$ $$\approx$$ $$2.475642903$$ $$L(\frac{5}{2})$$ 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 + (-1 - 1.73i)T + (-4 + 6.92i)T^{2}$$
7 $$1 + (-171.5 + 297. i)T^{2}$$
11 $$1 + (5 + 8.66i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + (-40 + 69.2i)T + (-1.09e3 - 1.90e3i)T^{2}$$
17 $$1 - 7T + 4.91e3T^{2}$$
19 $$1 + 113T + 6.85e3T^{2}$$
23 $$1 + (-40.5 + 70.1i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (-110 - 190. i)T + (-1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + (-94.5 + 163. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 - 170T + 5.06e4T^{2}$$
41 $$1 + (-65 + 112. i)T + (-3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (5 + 8.66i)T + (-3.97e4 + 6.88e4i)T^{2}$$
47 $$1 + (80 + 138. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 - 631T + 1.48e5T^{2}$$
59 $$1 + (-280 + 484. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (114.5 + 198. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (375 - 649. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 890T + 3.57e5T^{2}$$
73 $$1 + 890T + 3.89e5T^{2}$$
79 $$1 + (-13.5 - 23.3i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + (214.5 + 371. i)T + (-2.85e5 + 4.95e5i)T^{2}$$
89 $$1 + 750T + 7.04e5T^{2}$$
97 $$1 + (-740 - 1.28e3i)T + (-4.56e5 + 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$