# Properties

 Label 2-825-15.2-c1-0-11 Degree $2$ Conductor $825$ Sign $-0.229 - 0.973i$ Analytic cond. $6.58765$ Root an. cond. $2.56664$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.22 + 1.22i)2-s + (−1.22 + 1.22i)3-s − 0.999i·4-s − 2.99i·6-s + (−2.44 − 2.44i)7-s + (−1.22 − 1.22i)8-s − 2.99i·9-s − i·11-s + (1.22 + 1.22i)12-s + (2.44 − 2.44i)13-s + 5.99·14-s + 5·16-s + (−4.89 + 4.89i)17-s + (3.67 + 3.67i)18-s + 2i·19-s + ⋯
 L(s)  = 1 + (−0.866 + 0.866i)2-s + (−0.707 + 0.707i)3-s − 0.499i·4-s − 1.22i·6-s + (−0.925 − 0.925i)7-s + (−0.433 − 0.433i)8-s − 0.999i·9-s − 0.301i·11-s + (0.353 + 0.353i)12-s + (0.679 − 0.679i)13-s + 1.60·14-s + 1.25·16-s + (−1.18 + 1.18i)17-s + (0.866 + 0.866i)18-s + 0.458i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$825$$    =    $$3 \cdot 5^{2} \cdot 11$$ Sign: $-0.229 - 0.973i$ Analytic conductor: $$6.58765$$ Root analytic conductor: $$2.56664$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{825} (782, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 825,\ (\ :1/2),\ -0.229 - 0.973i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.328579 + 0.415178i$$ $$L(\frac12)$$ $$\approx$$ $$0.328579 + 0.415178i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (1.22 - 1.22i)T$$
5 $$1$$
11 $$1 + iT$$
good2 $$1 + (1.22 - 1.22i)T - 2iT^{2}$$
7 $$1 + (2.44 + 2.44i)T + 7iT^{2}$$
13 $$1 + (-2.44 + 2.44i)T - 13iT^{2}$$
17 $$1 + (4.89 - 4.89i)T - 17iT^{2}$$
19 $$1 - 2iT - 19T^{2}$$
23 $$1 + (-4.89 - 4.89i)T + 23iT^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 - 4T + 31T^{2}$$
37 $$1 + 37iT^{2}$$
41 $$1 - 6iT - 41T^{2}$$
43 $$1 + (-7.34 + 7.34i)T - 43iT^{2}$$
47 $$1 + (-4.89 + 4.89i)T - 47iT^{2}$$
53 $$1 + (-4.89 - 4.89i)T + 53iT^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 2T + 61T^{2}$$
67 $$1 + (2.44 + 2.44i)T + 67iT^{2}$$
71 $$1 + 12iT - 71T^{2}$$
73 $$1 + (2.44 - 2.44i)T - 73iT^{2}$$
79 $$1 - 10iT - 79T^{2}$$
83 $$1 + (-7.34 - 7.34i)T + 83iT^{2}$$
89 $$1 - 12T + 89T^{2}$$
97 $$1 + (-9.79 - 9.79i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$