Properties

 Label 2-15e2-5.4-c3-0-5 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 + i·2-s + 7·4-s + 24i·7-s + 15i·8-s − 52·11-s + 22i·13-s − 24·14-s + 41·16-s − 14i·17-s + 20·19-s − 52i·22-s + 168i·23-s − 22·26-s + 168i·28-s + 230·29-s + ⋯
 L(s)  = 1 + 0.353i·2-s + 0.875·4-s + 1.29i·7-s + 0.662i·8-s − 1.42·11-s + 0.469i·13-s − 0.458·14-s + 0.640·16-s − 0.199i·17-s + 0.241·19-s − 0.503i·22-s + 1.52i·23-s − 0.165·26-s + 1.13i·28-s + 1.47·29-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

 $$L(2)$$ $$\approx$$ $$0.928021 + 1.50157i$$ $$L(\frac12)$$ $$\approx$$ $$0.928021 + 1.50157i$$ $$L(\frac{5}{2})$$ 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 - iT - 8T^{2}$$
7 $$1 - 24iT - 343T^{2}$$
11 $$1 + 52T + 1.33e3T^{2}$$
13 $$1 - 22iT - 2.19e3T^{2}$$
17 $$1 + 14iT - 4.91e3T^{2}$$
19 $$1 - 20T + 6.85e3T^{2}$$
23 $$1 - 168iT - 1.21e4T^{2}$$
29 $$1 - 230T + 2.43e4T^{2}$$
31 $$1 + 288T + 2.97e4T^{2}$$
37 $$1 - 34iT - 5.06e4T^{2}$$
41 $$1 + 122T + 6.89e4T^{2}$$
43 $$1 + 188iT - 7.95e4T^{2}$$
47 $$1 - 256iT - 1.03e5T^{2}$$
53 $$1 - 338iT - 1.48e5T^{2}$$
59 $$1 - 100T + 2.05e5T^{2}$$
61 $$1 - 742T + 2.26e5T^{2}$$
67 $$1 - 84iT - 3.00e5T^{2}$$
71 $$1 - 328T + 3.57e5T^{2}$$
73 $$1 + 38iT - 3.89e5T^{2}$$
79 $$1 - 240T + 4.93e5T^{2}$$
83 $$1 + 1.21e3iT - 5.71e5T^{2}$$
89 $$1 - 330T + 7.04e5T^{2}$$
97 $$1 + 866iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$