# Properties

 Label 2-975-195.2-c0-0-0 Degree $2$ Conductor $975$ Sign $-0.913 - 0.406i$ 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 + (−0.965 − 0.258i)3-s + (−0.5 + 0.866i)4-s + (−0.448 − 0.258i)7-s + (0.866 + 0.499i)9-s + (0.707 − 0.707i)12-s + (−0.707 + 0.707i)13-s + (−0.499 − 0.866i)16-s + (−1.86 + 0.5i)19-s + (0.366 + 0.366i)21-s + (−0.707 − 0.707i)27-s + (0.448 − 0.258i)28-s + (−1 + i)31-s + (−0.866 + 0.5i)36-s + (−1.22 + 0.707i)37-s + (0.866 − 0.500i)39-s + ⋯
 L(s)  = 1 + (−0.965 − 0.258i)3-s + (−0.5 + 0.866i)4-s + (−0.448 − 0.258i)7-s + (0.866 + 0.499i)9-s + (0.707 − 0.707i)12-s + (−0.707 + 0.707i)13-s + (−0.499 − 0.866i)16-s + (−1.86 + 0.5i)19-s + (0.366 + 0.366i)21-s + (−0.707 − 0.707i)27-s + (0.448 − 0.258i)28-s + (−1 + i)31-s + (−0.866 + 0.5i)36-s + (−1.22 + 0.707i)37-s + (0.866 − 0.500i)39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$975$$    =    $$3 \cdot 5^{2} \cdot 13$$ Sign: $-0.913 - 0.406i$ Analytic conductor: $$0.486588$$ Root analytic conductor: $$0.697558$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{975} (782, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 975,\ (\ :0),\ -0.913 - 0.406i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.2224220517$$ $$L(\frac12)$$ $$\approx$$ $$0.2224220517$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (0.965 + 0.258i)T$$
5 $$1$$
13 $$1 + (0.707 - 0.707i)T$$
good2 $$1 + (0.5 - 0.866i)T^{2}$$
7 $$1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2}$$
11 $$1 + (0.866 + 0.5i)T^{2}$$
17 $$1 + (0.866 - 0.5i)T^{2}$$
19 $$1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2}$$
23 $$1 + (0.866 + 0.5i)T^{2}$$
29 $$1 + (-0.5 + 0.866i)T^{2}$$
31 $$1 + (1 - i)T - iT^{2}$$
37 $$1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}$$
41 $$1 + (-0.866 - 0.5i)T^{2}$$
43 $$1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2}$$
47 $$1 + T^{2}$$
53 $$1 + iT^{2}$$
59 $$1 + (0.866 - 0.5i)T^{2}$$
61 $$1 + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2}$$
71 $$1 + (0.866 - 0.5i)T^{2}$$
73 $$1 - 1.93T + T^{2}$$
79 $$1 + 1.73iT - T^{2}$$
83 $$1 + T^{2}$$
89 $$1 + (-0.866 - 0.5i)T^{2}$$
97 $$1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$