Properties

 Label 2-2475-5.4-c1-0-44 Degree $2$ Conductor $2475$ Sign $-0.447 + 0.894i$ Analytic cond. $19.7629$ Root an. cond. $4.44555$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 1.73i·2-s − 0.999·4-s + 2i·7-s − 1.73i·8-s + 11-s + 1.46i·13-s + 3.46·14-s − 5·16-s + 1.46·19-s − 1.73i·22-s − 6.92i·23-s + 2.53·26-s − 1.99i·28-s + 3.46·29-s + 2.92·31-s + 5.19i·32-s + ⋯
 L(s)  = 1 − 1.22i·2-s − 0.499·4-s + 0.755i·7-s − 0.612i·8-s + 0.301·11-s + 0.406i·13-s + 0.925·14-s − 1.25·16-s + 0.335·19-s − 0.369i·22-s − 1.44i·23-s + 0.497·26-s − 0.377i·28-s + 0.643·29-s + 0.525·31-s + 0.918i·32-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$2475$$    =    $$3^{2} \cdot 5^{2} \cdot 11$$ Sign: $-0.447 + 0.894i$ Analytic conductor: $$19.7629$$ Root analytic conductor: $$4.44555$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2475} (199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2475,\ (\ :1/2),\ -0.447 + 0.894i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.884876107$$ $$L(\frac12)$$ $$\approx$$ $$1.884876107$$ $$L(\frac{3}{2})$$ 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$$
11 $$1 - T$$
good2 $$1 + 1.73iT - 2T^{2}$$
7 $$1 - 2iT - 7T^{2}$$
13 $$1 - 1.46iT - 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 - 1.46T + 19T^{2}$$
23 $$1 + 6.92iT - 23T^{2}$$
29 $$1 - 3.46T + 29T^{2}$$
31 $$1 - 2.92T + 31T^{2}$$
37 $$1 - 8.92iT - 37T^{2}$$
41 $$1 - 3.46T + 41T^{2}$$
43 $$1 + 8.92iT - 43T^{2}$$
47 $$1 + 6.92iT - 47T^{2}$$
53 $$1 + 12.9iT - 53T^{2}$$
59 $$1 - 6.92T + 59T^{2}$$
61 $$1 - 2T + 61T^{2}$$
67 $$1 - 8iT - 67T^{2}$$
71 $$1 - 13.8T + 71T^{2}$$
73 $$1 + 12.3iT - 73T^{2}$$
79 $$1 - 13.4T + 79T^{2}$$
83 $$1 - 15.4iT - 83T^{2}$$
89 $$1 + 12.9T + 89T^{2}$$
97 $$1 + 10iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$