# Properties

 Degree $2$ Conductor $1225$ Sign $-0.967 + 0.254i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.448 − 1.67i)2-s + (−1.73 − 1.00i)4-s + (−1.22 + 1.22i)8-s + (0.866 − 0.5i)9-s + (0.5 − 0.866i)11-s + (0.500 + 0.866i)16-s + (−0.448 − 1.67i)18-s + (−1.22 − 1.22i)22-s + (−1.67 − 0.448i)23-s + i·29-s − 2·36-s + (−0.448 + 1.67i)37-s + (1.22 − 1.22i)43-s + (−1.73 + 0.999i)44-s + (−1.50 + 2.59i)46-s + ⋯
 L(s)  = 1 + (0.448 − 1.67i)2-s + (−1.73 − 1.00i)4-s + (−1.22 + 1.22i)8-s + (0.866 − 0.5i)9-s + (0.5 − 0.866i)11-s + (0.500 + 0.866i)16-s + (−0.448 − 1.67i)18-s + (−1.22 − 1.22i)22-s + (−1.67 − 0.448i)23-s + i·29-s − 2·36-s + (−0.448 + 1.67i)37-s + (1.22 − 1.22i)43-s + (−1.73 + 0.999i)44-s + (−1.50 + 2.59i)46-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1225$$    =    $$5^{2} \cdot 7^{2}$$ Sign: $-0.967 + 0.254i$ Motivic weight: $$0$$ Character: $\chi_{1225} (557, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1225,\ (\ :0),\ -0.967 + 0.254i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.253404666$$ $$L(\frac12)$$ $$\approx$$ $$1.253404666$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
7 $$1$$
good2 $$1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2}$$
3 $$1 + (-0.866 + 0.5i)T^{2}$$
11 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
13 $$1 - iT^{2}$$
17 $$1 + (0.866 - 0.5i)T^{2}$$
19 $$1 + (0.5 - 0.866i)T^{2}$$
23 $$1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2}$$
29 $$1 - iT - T^{2}$$
31 $$1 + (-0.5 - 0.866i)T^{2}$$
37 $$1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2}$$
41 $$1 + T^{2}$$
43 $$1 + (-1.22 + 1.22i)T - iT^{2}$$
47 $$1 + (-0.866 - 0.5i)T^{2}$$
53 $$1 + (-0.866 + 0.5i)T^{2}$$
59 $$1 + (0.5 + 0.866i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T^{2}$$
67 $$1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2}$$
71 $$1 - T + T^{2}$$
73 $$1 + (-0.866 + 0.5i)T^{2}$$
79 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + (0.5 - 0.866i)T^{2}$$
97 $$1 + iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$