# Properties

 Label 2-1008-7.3-c2-0-13 Degree $2$ Conductor $1008$ Sign $0.124 - 0.992i$ Analytic cond. $27.4660$ Root an. cond. $5.24080$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.24 − 0.717i)5-s + (−1.74 + 6.77i)7-s + (−3 + 5.19i)11-s − 21.3i·13-s + (7.75 + 4.47i)17-s + (6.25 − 3.61i)19-s + (18.7 + 32.4i)23-s + (−11.4 + 19.8i)25-s + 33.9·29-s + (−38.2 − 22.0i)31-s + (2.69 + 9.67i)35-s + (13.9 + 24.2i)37-s + 54.8i·41-s + 1.48·43-s + (−37.2 + 21.5i)47-s + ⋯
 L(s)  = 1 + (0.248 − 0.143i)5-s + (−0.248 + 0.968i)7-s + (−0.272 + 0.472i)11-s − 1.64i·13-s + (0.456 + 0.263i)17-s + (0.329 − 0.190i)19-s + (0.814 + 1.41i)23-s + (−0.458 + 0.794i)25-s + 1.17·29-s + (−1.23 − 0.711i)31-s + (0.0770 + 0.276i)35-s + (0.377 + 0.654i)37-s + 1.33i·41-s + 0.0345·43-s + (−0.792 + 0.457i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ Sign: $0.124 - 0.992i$ Analytic conductor: $$27.4660$$ Root analytic conductor: $$5.24080$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1008} (577, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1008,\ (\ :1),\ 0.124 - 0.992i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.566710130$$ $$L(\frac12)$$ $$\approx$$ $$1.566710130$$ $$L(2)$$ 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$$
7 $$1 + (1.74 - 6.77i)T$$
good5 $$1 + (-1.24 + 0.717i)T + (12.5 - 21.6i)T^{2}$$
11 $$1 + (3 - 5.19i)T + (-60.5 - 104. i)T^{2}$$
13 $$1 + 21.3iT - 169T^{2}$$
17 $$1 + (-7.75 - 4.47i)T + (144.5 + 250. i)T^{2}$$
19 $$1 + (-6.25 + 3.61i)T + (180.5 - 312. i)T^{2}$$
23 $$1 + (-18.7 - 32.4i)T + (-264.5 + 458. i)T^{2}$$
29 $$1 - 33.9T + 841T^{2}$$
31 $$1 + (38.2 + 22.0i)T + (480.5 + 832. i)T^{2}$$
37 $$1 + (-13.9 - 24.2i)T + (-684.5 + 1.18e3i)T^{2}$$
41 $$1 - 54.8iT - 1.68e3T^{2}$$
43 $$1 - 1.48T + 1.84e3T^{2}$$
47 $$1 + (37.2 - 21.5i)T + (1.10e3 - 1.91e3i)T^{2}$$
53 $$1 + (42.7 - 74.0i)T + (-1.40e3 - 2.43e3i)T^{2}$$
59 $$1 + (35.6 + 20.6i)T + (1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (1.02 - 0.594i)T + (1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (-2.19 + 3.80i)T + (-2.24e3 - 3.88e3i)T^{2}$$
71 $$1 - 137.T + 5.04e3T^{2}$$
73 $$1 + (-68.3 - 39.4i)T + (2.66e3 + 4.61e3i)T^{2}$$
79 $$1 + (-49.1 - 85.1i)T + (-3.12e3 + 5.40e3i)T^{2}$$
83 $$1 - 110. iT - 6.88e3T^{2}$$
89 $$1 + (-18 + 10.3i)T + (3.96e3 - 6.85e3i)T^{2}$$
97 $$1 + 10.9iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$