# Properties

 Label 2-2925-5.4-c1-0-62 Degree $2$ Conductor $2925$ Sign $0.894 + 0.447i$ Analytic cond. $23.3562$ Root an. cond. $4.83282$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·4-s − i·7-s + 3·11-s − i·13-s + 4·16-s + 3i·17-s + 4·19-s − 9i·23-s − 2i·28-s − 6·29-s + 2·31-s − i·37-s + 3·41-s − 2i·43-s + 6·44-s + ⋯
 L(s)  = 1 + 4-s − 0.377i·7-s + 0.904·11-s − 0.277i·13-s + 16-s + 0.727i·17-s + 0.917·19-s − 1.87i·23-s − 0.377i·28-s − 1.11·29-s + 0.359·31-s − 0.164i·37-s + 0.468·41-s − 0.304i·43-s + 0.904·44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2925$$    =    $$3^{2} \cdot 5^{2} \cdot 13$$ Sign: $0.894 + 0.447i$ Analytic conductor: $$23.3562$$ Root analytic conductor: $$4.83282$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2925} (2224, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2925,\ (\ :1/2),\ 0.894 + 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.645692576$$ $$L(\frac12)$$ $$\approx$$ $$2.645692576$$ $$L(\frac{3}{2})$$ 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$$
13 $$1 + iT$$
good2 $$1 - 2T^{2}$$
7 $$1 + iT - 7T^{2}$$
11 $$1 - 3T + 11T^{2}$$
17 $$1 - 3iT - 17T^{2}$$
19 $$1 - 4T + 19T^{2}$$
23 $$1 + 9iT - 23T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 - 2T + 31T^{2}$$
37 $$1 + iT - 37T^{2}$$
41 $$1 - 3T + 41T^{2}$$
43 $$1 + 2iT - 43T^{2}$$
47 $$1 - 6iT - 47T^{2}$$
53 $$1 - 9iT - 53T^{2}$$
59 $$1 + 12T + 59T^{2}$$
61 $$1 - 5T + 61T^{2}$$
67 $$1 + 4iT - 67T^{2}$$
71 $$1 + 9T + 71T^{2}$$
73 $$1 + 14iT - 73T^{2}$$
79 $$1 - 7T + 79T^{2}$$
83 $$1 - 83T^{2}$$
89 $$1 - 15T + 89T^{2}$$
97 $$1 - 5iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$