# Properties

 Label 2-15e2-15.8-c1-0-5 Degree $2$ Conductor $225$ Sign $-0.391 + 0.920i$ Analytic cond. $1.79663$ Root an. cond. $1.34038$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 − 0.707i)2-s − 0.999i·4-s + (2 − 2i)7-s + (−2.12 + 2.12i)8-s − 2.82i·11-s + (−1 − i)13-s − 2.82·14-s + 1.00·16-s + (−2.82 − 2.82i)17-s + (−2.00 + 2.00i)22-s + (2.82 − 2.82i)23-s + 1.41i·26-s + (−1.99 − 1.99i)28-s + 4.24·29-s − 4·31-s + (3.53 + 3.53i)32-s + ⋯
 L(s)  = 1 + (−0.499 − 0.499i)2-s − 0.499i·4-s + (0.755 − 0.755i)7-s + (−0.750 + 0.750i)8-s − 0.852i·11-s + (−0.277 − 0.277i)13-s − 0.755·14-s + 0.250·16-s + (−0.685 − 0.685i)17-s + (−0.426 + 0.426i)22-s + (0.589 − 0.589i)23-s + 0.277i·26-s + (−0.377 − 0.377i)28-s + 0.787·29-s − 0.718·31-s + (0.624 + 0.624i)32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $-0.391 + 0.920i$ Analytic conductor: $$1.79663$$ Root analytic conductor: $$1.34038$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{225} (143, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 225,\ (\ :1/2),\ -0.391 + 0.920i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.502559 - 0.759545i$$ $$L(\frac12)$$ $$\approx$$ $$0.502559 - 0.759545i$$ $$L(\frac{3}{2})$$ 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 + (0.707 + 0.707i)T + 2iT^{2}$$
7 $$1 + (-2 + 2i)T - 7iT^{2}$$
11 $$1 + 2.82iT - 11T^{2}$$
13 $$1 + (1 + i)T + 13iT^{2}$$
17 $$1 + (2.82 + 2.82i)T + 17iT^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 + (-2.82 + 2.82i)T - 23iT^{2}$$
29 $$1 - 4.24T + 29T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 + (1 - i)T - 37iT^{2}$$
41 $$1 - 1.41iT - 41T^{2}$$
43 $$1 + (-8 - 8i)T + 43iT^{2}$$
47 $$1 + (-5.65 - 5.65i)T + 47iT^{2}$$
53 $$1 + (-2.82 + 2.82i)T - 53iT^{2}$$
59 $$1 - 8.48T + 59T^{2}$$
61 $$1 - 8T + 61T^{2}$$
67 $$1 + (4 - 4i)T - 67iT^{2}$$
71 $$1 - 5.65iT - 71T^{2}$$
73 $$1 + (1 + i)T + 73iT^{2}$$
79 $$1 - 12iT - 79T^{2}$$
83 $$1 + (-2.82 + 2.82i)T - 83iT^{2}$$
89 $$1 + 12.7T + 89T^{2}$$
97 $$1 + (-11 + 11i)T - 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$