# Properties

 Label 2-912-57.2-c1-0-22 Degree $2$ Conductor $912$ Sign $0.962 + 0.271i$ 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 + (1.69 − 0.355i)3-s + (−0.882 − 2.42i)5-s + (1.58 + 2.74i)7-s + (2.74 − 1.20i)9-s + (2.16 + 1.25i)11-s + (2.71 + 3.24i)13-s + (−2.35 − 3.79i)15-s + (1.32 + 0.233i)17-s + (−3.14 + 3.01i)19-s + (3.65 + 4.08i)21-s + (1.30 − 3.58i)23-s + (−1.27 + 1.06i)25-s + (4.23 − 3.01i)27-s + (−1.32 − 7.49i)29-s + (−6.89 + 3.97i)31-s + ⋯
 L(s)  = 1 + (0.978 − 0.205i)3-s + (−0.394 − 1.08i)5-s + (0.598 + 1.03i)7-s + (0.915 − 0.401i)9-s + (0.653 + 0.377i)11-s + (0.754 + 0.898i)13-s + (−0.608 − 0.980i)15-s + (0.320 + 0.0565i)17-s + (−0.722 + 0.691i)19-s + (0.798 + 0.892i)21-s + (0.272 − 0.747i)23-s + (−0.254 + 0.213i)25-s + (0.814 − 0.580i)27-s + (−0.245 − 1.39i)29-s + (−1.23 + 0.714i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$912$$    =    $$2^{4} \cdot 3 \cdot 19$$ Sign: $0.962 + 0.271i$ Analytic conductor: $$7.28235$$ Root analytic conductor: $$2.69858$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{912} (401, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 912,\ (\ :1/2),\ 0.962 + 0.271i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.35671 - 0.325795i$$ $$L(\frac12)$$ $$\approx$$ $$2.35671 - 0.325795i$$ $$L(\frac{3}{2})$$ 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 + (-1.69 + 0.355i)T$$
19 $$1 + (3.14 - 3.01i)T$$
good5 $$1 + (0.882 + 2.42i)T + (-3.83 + 3.21i)T^{2}$$
7 $$1 + (-1.58 - 2.74i)T + (-3.5 + 6.06i)T^{2}$$
11 $$1 + (-2.16 - 1.25i)T + (5.5 + 9.52i)T^{2}$$
13 $$1 + (-2.71 - 3.24i)T + (-2.25 + 12.8i)T^{2}$$
17 $$1 + (-1.32 - 0.233i)T + (15.9 + 5.81i)T^{2}$$
23 $$1 + (-1.30 + 3.58i)T + (-17.6 - 14.7i)T^{2}$$
29 $$1 + (1.32 + 7.49i)T + (-27.2 + 9.91i)T^{2}$$
31 $$1 + (6.89 - 3.97i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 + 4.10iT - 37T^{2}$$
41 $$1 + (-4.95 - 4.16i)T + (7.11 + 40.3i)T^{2}$$
43 $$1 + (-11.7 + 4.27i)T + (32.9 - 27.6i)T^{2}$$
47 $$1 + (6.16 - 1.08i)T + (44.1 - 16.0i)T^{2}$$
53 $$1 + (3.46 + 1.26i)T + (40.6 + 34.0i)T^{2}$$
59 $$1 + (-1.54 + 8.75i)T + (-55.4 - 20.1i)T^{2}$$
61 $$1 + (0.133 + 0.0485i)T + (46.7 + 39.2i)T^{2}$$
67 $$1 + (-4.48 + 0.791i)T + (62.9 - 22.9i)T^{2}$$
71 $$1 + (8.59 - 3.12i)T + (54.3 - 45.6i)T^{2}$$
73 $$1 + (1.67 + 1.40i)T + (12.6 + 71.8i)T^{2}$$
79 $$1 + (6.41 - 7.64i)T + (-13.7 - 77.7i)T^{2}$$
83 $$1 + (12.3 - 7.11i)T + (41.5 - 71.8i)T^{2}$$
89 $$1 + (12.7 - 10.6i)T + (15.4 - 87.6i)T^{2}$$
97 $$1 + (-0.538 - 0.0949i)T + (91.1 + 33.1i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$