# Properties

 Degree $2$ Conductor $525$ Sign $0.447 + 0.894i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# 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 − 36·11-s + 3i·12-s − 34i·13-s + 21·14-s − 71·16-s − 42i·17-s − 27i·18-s + 124·19-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 − 0.986·11-s + 0.0721i·12-s − 0.725i·13-s + 0.400·14-s − 1.10·16-s − 0.599i·17-s − 0.353i·18-s + 1.49·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{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: $$525$$    =    $$3 \cdot 5^{2} \cdot 7$$ Sign: $0.447 + 0.894i$ Motivic weight: $$3$$ Character: $\chi_{525} (274, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 525,\ (\ :3/2),\ 0.447 + 0.894i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.199350181$$ $$L(\frac12)$$ $$\approx$$ $$1.199350181$$ $$L(\frac{5}{2})$$ 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$$
7 $$1 + 7iT$$
good2 $$1 - 3iT - 8T^{2}$$
11 $$1 + 36T + 1.33e3T^{2}$$
13 $$1 + 34iT - 2.19e3T^{2}$$
17 $$1 + 42iT - 4.91e3T^{2}$$
19 $$1 - 124T + 6.85e3T^{2}$$
23 $$1 - 1.21e4T^{2}$$
29 $$1 + 102T + 2.43e4T^{2}$$
31 $$1 + 160T + 2.97e4T^{2}$$
37 $$1 + 398iT - 5.06e4T^{2}$$
41 $$1 + 318T + 6.89e4T^{2}$$
43 $$1 + 268iT - 7.95e4T^{2}$$
47 $$1 + 240iT - 1.03e5T^{2}$$
53 $$1 + 498iT - 1.48e5T^{2}$$
59 $$1 - 132T + 2.05e5T^{2}$$
61 $$1 - 398T + 2.26e5T^{2}$$
67 $$1 + 92iT - 3.00e5T^{2}$$
71 $$1 + 720T + 3.57e5T^{2}$$
73 $$1 + 502iT - 3.89e5T^{2}$$
79 $$1 - 1.02e3T + 4.93e5T^{2}$$
83 $$1 + 204iT - 5.71e5T^{2}$$
89 $$1 + 354T + 7.04e5T^{2}$$
97 $$1 - 286iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$