# Properties

 Label 2-1925-5.4-c1-0-75 Degree $2$ Conductor $1925$ Sign $-0.894 + 0.447i$ Analytic cond. $15.3712$ Root an. cond. $3.92061$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3i·3-s + 2·4-s + i·7-s − 6·9-s − 11-s − 6i·12-s − 4i·13-s + 4·16-s − 2i·17-s + 6·19-s + 3·21-s − 5i·23-s + 9i·27-s + 2i·28-s − 10·29-s + ⋯
 L(s)  = 1 − 1.73i·3-s + 4-s + 0.377i·7-s − 2·9-s − 0.301·11-s − 1.73i·12-s − 1.10i·13-s + 16-s − 0.485i·17-s + 1.37·19-s + 0.654·21-s − 1.04i·23-s + 1.73i·27-s + 0.377i·28-s − 1.85·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{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: $$1925$$    =    $$5^{2} \cdot 7 \cdot 11$$ Sign: $-0.894 + 0.447i$ Analytic conductor: $$15.3712$$ Root analytic conductor: $$3.92061$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1925} (1849, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1925,\ (\ :1/2),\ -0.894 + 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.906419955$$ $$L(\frac12)$$ $$\approx$$ $$1.906419955$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
7 $$1 - iT$$
11 $$1 + T$$
good2 $$1 - 2T^{2}$$
3 $$1 + 3iT - 3T^{2}$$
13 $$1 + 4iT - 13T^{2}$$
17 $$1 + 2iT - 17T^{2}$$
19 $$1 - 6T + 19T^{2}$$
23 $$1 + 5iT - 23T^{2}$$
29 $$1 + 10T + 29T^{2}$$
31 $$1 - T + 31T^{2}$$
37 $$1 - 5iT - 37T^{2}$$
41 $$1 + 2T + 41T^{2}$$
43 $$1 + 8iT - 43T^{2}$$
47 $$1 + 8iT - 47T^{2}$$
53 $$1 + 6iT - 53T^{2}$$
59 $$1 + 3T + 59T^{2}$$
61 $$1 + 2T + 61T^{2}$$
67 $$1 - 3iT - 67T^{2}$$
71 $$1 - T + 71T^{2}$$
73 $$1 - 10iT - 73T^{2}$$
79 $$1 + 6T + 79T^{2}$$
83 $$1 - 12iT - 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}$$