# Properties

 Label 2-1008-21.17-c3-0-20 Degree $2$ Conductor $1008$ Sign $0.330 - 0.943i$ Analytic cond. $59.4739$ Root an. cond. $7.71193$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.35 + 2.35i)5-s + (8.74 + 16.3i)7-s + (8.39 + 4.84i)11-s + 67.7i·13-s + (50.1 − 86.7i)17-s + (59.6 − 34.4i)19-s + (126. − 73.3i)23-s + (58.8 − 101. i)25-s + 284. i·29-s + (−197. − 113. i)31-s + (−26.5 + 42.7i)35-s + (150. + 261. i)37-s − 232.·41-s − 173.·43-s + (191. + 331. i)47-s + ⋯
 L(s)  = 1 + (0.121 + 0.210i)5-s + (0.471 + 0.881i)7-s + (0.229 + 0.132i)11-s + 1.44i·13-s + (0.714 − 1.23i)17-s + (0.720 − 0.416i)19-s + (1.15 − 0.664i)23-s + (0.470 − 0.814i)25-s + 1.82i·29-s + (−1.14 − 0.659i)31-s + (−0.128 + 0.206i)35-s + (0.669 + 1.16i)37-s − 0.885·41-s − 0.613·43-s + (0.594 + 1.02i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ Sign: $0.330 - 0.943i$ Analytic conductor: $$59.4739$$ Root analytic conductor: $$7.71193$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1008} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1008,\ (\ :3/2),\ 0.330 - 0.943i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.359872447$$ $$L(\frac12)$$ $$\approx$$ $$2.359872447$$ $$L(\frac{5}{2})$$ 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 + (-8.74 - 16.3i)T$$
good5 $$1 + (-1.35 - 2.35i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (-8.39 - 4.84i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 - 67.7iT - 2.19e3T^{2}$$
17 $$1 + (-50.1 + 86.7i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-59.6 + 34.4i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-126. + 73.3i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 284. iT - 2.43e4T^{2}$$
31 $$1 + (197. + 113. i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (-150. - 261. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 232.T + 6.89e4T^{2}$$
43 $$1 + 173.T + 7.95e4T^{2}$$
47 $$1 + (-191. - 331. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (22.2 + 12.8i)T + (7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-371. + 644. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (343. - 198. i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-293. + 508. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 501. iT - 3.57e5T^{2}$$
73 $$1 + (-616. - 356. i)T + (1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (466. + 807. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 - 837.T + 5.71e5T^{2}$$
89 $$1 + (-442. - 766. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 1.17e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$