# Properties

 Label 2-1008-21.17-c3-0-12 Degree $2$ Conductor $1008$ Sign $-0.548 - 0.836i$ 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 + (2.08 + 3.60i)5-s + (18.2 + 2.97i)7-s + (33.1 + 19.1i)11-s + 49.6i·13-s + (−7.65 + 13.2i)17-s + (−122. + 70.9i)19-s + (−136. + 78.9i)23-s + (53.8 − 93.2i)25-s + 204. i·29-s + (90.5 + 52.2i)31-s + (27.3 + 72.1i)35-s + (−194. − 336. i)37-s + 325.·41-s − 191.·43-s + (−249. − 432. i)47-s + ⋯
 L(s)  = 1 + (0.186 + 0.322i)5-s + (0.986 + 0.160i)7-s + (0.909 + 0.524i)11-s + 1.05i·13-s + (−0.109 + 0.189i)17-s + (−1.48 + 0.856i)19-s + (−1.24 + 0.715i)23-s + (0.430 − 0.745i)25-s + 1.31i·29-s + (0.524 + 0.302i)31-s + (0.131 + 0.348i)35-s + (−0.862 − 1.49i)37-s + 1.23·41-s − 0.678·43-s + (−0.775 − 1.34i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ Sign: $-0.548 - 0.836i$ 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: $$(2,\ 1008,\ (\ :3/2),\ -0.548 - 0.836i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.805059973$$ $$L(\frac12)$$ $$\approx$$ $$1.805059973$$ $$L(\frac{5}{2})$$ 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 + (-18.2 - 2.97i)T$$
good5 $$1 + (-2.08 - 3.60i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (-33.1 - 19.1i)T + (665.5 + 1.15e3i)T^{2}$$
13 $$1 - 49.6iT - 2.19e3T^{2}$$
17 $$1 + (7.65 - 13.2i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (122. - 70.9i)T + (3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (136. - 78.9i)T + (6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 204. iT - 2.43e4T^{2}$$
31 $$1 + (-90.5 - 52.2i)T + (1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (194. + 336. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 - 325.T + 6.89e4T^{2}$$
43 $$1 + 191.T + 7.95e4T^{2}$$
47 $$1 + (249. + 432. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (-37.8 - 21.8i)T + (7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-86.5 + 149. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-208. + 120. i)T + (1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (440. - 763. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 1.01e3iT - 3.57e5T^{2}$$
73 $$1 + (361. + 208. i)T + (1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (237. + 411. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + 652.T + 5.71e5T^{2}$$
89 $$1 + (-298. - 517. i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 + 1.77e3iT - 9.12e5T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.957557072656829274558725731788, −8.867876446076707233718586923541, −8.398000657288273016673188308502, −7.24022616142347307192433419230, −6.55069745265369616735061305175, −5.62928769952321773080113632540, −4.44440954245558136558158125491, −3.85795304564655784111381832071, −2.16335438704687857609837456043, −1.56366115121976316850016375870, 0.43089977143013683550161206752, 1.57613939628257034894626680665, 2.77526351252379080179330911720, 4.14028566551149072385370659536, 4.80077841066117245298179660995, 5.91442772979557206098967994585, 6.62971387915417726839985174072, 7.891970124577835785855130331896, 8.378576848009440667831427976426, 9.191214743268108663768911833913