Properties

 Label 2-15e2-5.4-c3-0-11 Degree $2$ Conductor $225$ Sign $0.447 - 0.894i$ Analytic cond. $13.2754$ Root an. cond. $3.64354$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 5i·2-s − 17·4-s − 30i·7-s − 45i·8-s + 50·11-s + 20i·13-s + 150·14-s + 89·16-s − 10i·17-s + 44·19-s + 250i·22-s − 120i·23-s − 100·26-s + 510i·28-s + 50·29-s + ⋯
 L(s)  = 1 + 1.76i·2-s − 2.12·4-s − 1.61i·7-s − 1.98i·8-s + 1.37·11-s + 0.426i·13-s + 2.86·14-s + 1.39·16-s − 0.142i·17-s + 0.531·19-s + 2.42i·22-s − 1.08i·23-s − 0.754·26-s + 3.44i·28-s + 0.320·29-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $0.447 - 0.894i$ Analytic conductor: $$13.2754$$ Root analytic conductor: $$3.64354$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{225} (199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 225,\ (\ :3/2),\ 0.447 - 0.894i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$1.32233 + 0.817247i$$ $$L(\frac12)$$ $$\approx$$ $$1.32233 + 0.817247i$$ $$L(\frac{5}{2})$$ 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$$
good2 $$1 - 5iT - 8T^{2}$$
7 $$1 + 30iT - 343T^{2}$$
11 $$1 - 50T + 1.33e3T^{2}$$
13 $$1 - 20iT - 2.19e3T^{2}$$
17 $$1 + 10iT - 4.91e3T^{2}$$
19 $$1 - 44T + 6.85e3T^{2}$$
23 $$1 + 120iT - 1.21e4T^{2}$$
29 $$1 - 50T + 2.43e4T^{2}$$
31 $$1 - 108T + 2.97e4T^{2}$$
37 $$1 + 40iT - 5.06e4T^{2}$$
41 $$1 - 400T + 6.89e4T^{2}$$
43 $$1 + 280iT - 7.95e4T^{2}$$
47 $$1 + 280iT - 1.03e5T^{2}$$
53 $$1 - 610iT - 1.48e5T^{2}$$
59 $$1 + 50T + 2.05e5T^{2}$$
61 $$1 + 518T + 2.26e5T^{2}$$
67 $$1 + 180iT - 3.00e5T^{2}$$
71 $$1 - 700T + 3.57e5T^{2}$$
73 $$1 - 410iT - 3.89e5T^{2}$$
79 $$1 - 516T + 4.93e5T^{2}$$
83 $$1 + 660iT - 5.71e5T^{2}$$
89 $$1 - 1.50e3T + 7.04e5T^{2}$$
97 $$1 + 1.63e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$