# Properties

 Label 2-975-195.83-c0-0-0 Degree $2$ Conductor $975$ Sign $0.661 - 0.749i$ Analytic cond. $0.486588$ Root an. cond. $0.697558$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 − 0.707i)3-s − 4-s − 1.41·7-s + 1.00i·9-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s + 16-s + (1 + i)19-s + (1.00 + 1.00i)21-s + (0.707 − 0.707i)27-s + 1.41·28-s + (−1 + i)31-s − 1.00i·36-s + 1.41·37-s − 1.00i·39-s + ⋯
 L(s)  = 1 + (−0.707 − 0.707i)3-s − 4-s − 1.41·7-s + 1.00i·9-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s + 16-s + (1 + i)19-s + (1.00 + 1.00i)21-s + (0.707 − 0.707i)27-s + 1.41·28-s + (−1 + i)31-s − 1.00i·36-s + 1.41·37-s − 1.00i·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$975$$    =    $$3 \cdot 5^{2} \cdot 13$$ Sign: $0.661 - 0.749i$ Analytic conductor: $$0.486588$$ Root analytic conductor: $$0.697558$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{975} (668, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 975,\ (\ :0),\ 0.661 - 0.749i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4410441409$$ $$L(\frac12)$$ $$\approx$$ $$0.4410441409$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (0.707 + 0.707i)T$$
5 $$1$$
13 $$1 + (-0.707 - 0.707i)T$$
good2 $$1 + T^{2}$$
7 $$1 + 1.41T + T^{2}$$
11 $$1 - iT^{2}$$
17 $$1 - iT^{2}$$
19 $$1 + (-1 - i)T + iT^{2}$$
23 $$1 + iT^{2}$$
29 $$1 + T^{2}$$
31 $$1 + (1 - i)T - iT^{2}$$
37 $$1 - 1.41T + T^{2}$$
41 $$1 + iT^{2}$$
43 $$1 + (1.41 - 1.41i)T - iT^{2}$$
47 $$1 - T^{2}$$
53 $$1 - iT^{2}$$
59 $$1 + iT^{2}$$
61 $$1 + T^{2}$$
67 $$1 - 1.41iT - T^{2}$$
71 $$1 + iT^{2}$$
73 $$1 - 1.41iT - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + iT^{2}$$
97 $$1 - 1.41iT - T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$