# Properties

 Label 2-825-5.4-c3-0-85 Degree $2$ Conductor $825$ Sign $-0.447 - 0.894i$ Analytic cond. $48.6765$ Root an. cond. $6.97686$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3i·2-s + 3i·3-s − 4-s + 9·6-s − 7i·7-s − 21i·8-s − 9·9-s + 11·11-s − 3i·12-s + 16i·13-s − 21·14-s − 71·16-s + 21i·17-s + 27i·18-s − 125·19-s + ⋯
 L(s)  = 1 − 1.06i·2-s + 0.577i·3-s − 0.125·4-s + 0.612·6-s − 0.377i·7-s − 0.928i·8-s − 0.333·9-s + 0.301·11-s − 0.0721i·12-s + 0.341i·13-s − 0.400·14-s − 1.10·16-s + 0.299i·17-s + 0.353i·18-s − 1.50·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{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: $$825$$    =    $$3 \cdot 5^{2} \cdot 11$$ Sign: $-0.447 - 0.894i$ Analytic conductor: $$48.6765$$ Root analytic conductor: $$6.97686$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{825} (199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 825,\ (\ :3/2),\ -0.447 - 0.894i)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 3iT$$
5 $$1$$
11 $$1 - 11T$$
good2 $$1 + 3iT - 8T^{2}$$
7 $$1 + 7iT - 343T^{2}$$
13 $$1 - 16iT - 2.19e3T^{2}$$
17 $$1 - 21iT - 4.91e3T^{2}$$
19 $$1 + 125T + 6.85e3T^{2}$$
23 $$1 - 81iT - 1.21e4T^{2}$$
29 $$1 + 186T + 2.43e4T^{2}$$
31 $$1 + 58T + 2.97e4T^{2}$$
37 $$1 + 253iT - 5.06e4T^{2}$$
41 $$1 - 63T + 6.89e4T^{2}$$
43 $$1 - 100iT - 7.95e4T^{2}$$
47 $$1 + 219iT - 1.03e5T^{2}$$
53 $$1 - 192iT - 1.48e5T^{2}$$
59 $$1 + 249T + 2.05e5T^{2}$$
61 $$1 + 64T + 2.26e5T^{2}$$
67 $$1 - 272iT - 3.00e5T^{2}$$
71 $$1 + 645T + 3.57e5T^{2}$$
73 $$1 - 112iT - 3.89e5T^{2}$$
79 $$1 + 509T + 4.93e5T^{2}$$
83 $$1 - 1.25e3iT - 5.71e5T^{2}$$
89 $$1 + 756T + 7.04e5T^{2}$$
97 $$1 - 839iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$