# Properties

 Label 2-912-76.27-c1-0-5 Degree $2$ Conductor $912$ Sign $-0.321 - 0.946i$ Analytic cond. $7.28235$ Root an. cond. $2.69858$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 + 0.866i)3-s + (1.33 − 2.31i)5-s + 3.93i·7-s + (−0.499 − 0.866i)9-s + 2.20i·11-s + (−3.60 + 2.07i)13-s + (1.33 + 2.31i)15-s + (−0.571 + 0.990i)17-s + (−4.24 + 0.990i)19-s + (−3.40 − 1.96i)21-s + (3.19 − 1.84i)23-s + (−1.07 − 1.85i)25-s + 0.999·27-s + (−2.10 + 1.21i)29-s + 3.67·31-s + ⋯
 L(s)  = 1 + (−0.288 + 0.499i)3-s + (0.597 − 1.03i)5-s + 1.48i·7-s + (−0.166 − 0.288i)9-s + 0.664i·11-s + (−0.998 + 0.576i)13-s + (0.345 + 0.597i)15-s + (−0.138 + 0.240i)17-s + (−0.973 + 0.227i)19-s + (−0.743 − 0.429i)21-s + (0.665 − 0.384i)23-s + (−0.214 − 0.371i)25-s + 0.192·27-s + (−0.390 + 0.225i)29-s + 0.659·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$912$$    =    $$2^{4} \cdot 3 \cdot 19$$ Sign: $-0.321 - 0.946i$ Analytic conductor: $$7.28235$$ Root analytic conductor: $$2.69858$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{912} (559, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 912,\ (\ :1/2),\ -0.321 - 0.946i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.672913 + 0.938983i$$ $$L(\frac12)$$ $$\approx$$ $$0.672913 + 0.938983i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (0.5 - 0.866i)T$$
19 $$1 + (4.24 - 0.990i)T$$
good5 $$1 + (-1.33 + 2.31i)T + (-2.5 - 4.33i)T^{2}$$
7 $$1 - 3.93iT - 7T^{2}$$
11 $$1 - 2.20iT - 11T^{2}$$
13 $$1 + (3.60 - 2.07i)T + (6.5 - 11.2i)T^{2}$$
17 $$1 + (0.571 - 0.990i)T + (-8.5 - 14.7i)T^{2}$$
23 $$1 + (-3.19 + 1.84i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + (2.10 - 1.21i)T + (14.5 - 25.1i)T^{2}$$
31 $$1 - 3.67T + 31T^{2}$$
37 $$1 - 10.0iT - 37T^{2}$$
41 $$1 + (-8.01 - 4.62i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (0.490 + 0.283i)T + (21.5 + 37.2i)T^{2}$$
47 $$1 + (10.6 - 6.13i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (-4.09 + 2.36i)T + (26.5 - 45.8i)T^{2}$$
59 $$1 + (1.33 - 2.31i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (6.41 + 11.1i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-1.73 - 3.00i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (6.24 - 10.8i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (-1.35 + 2.34i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (3.50 - 6.07i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 1.98iT - 83T^{2}$$
89 $$1 + (-12.9 + 7.46i)T + (44.5 - 77.0i)T^{2}$$
97 $$1 + (-2.91 - 1.68i)T + (48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$