# Properties

 Label 2-1008-21.17-c3-0-40 Degree $2$ Conductor $1008$ Sign $-0.605 + 0.795i$ Analytic cond. $59.4739$ Root an. cond. $7.71193$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## 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 & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ Sign: $-0.605 + 0.795i$ 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: $$(2,\ 1008,\ (\ :3/2),\ -0.605 + 0.795i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.011117892$$ $$L(\frac12)$$ $$\approx$$ $$1.011117892$$ $$L(\frac{5}{2})$$ 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

−9.008444096941772702943131041926, −8.496863261740025371385081449266, −7.80571381436408923620519535862, −6.74698522699245296605707695335, −5.79581036113470366136805086589, −4.78299487331070625229849817772, −4.17969319509129912928386439497, −2.69634698190700312418199569346, −1.68504876076647502987462779589, −0.25179396297026974895483157530, 1.28069831226899701971417002270, 2.53284169736768899842645197554, 3.53136796419573901432647291050, 4.95917627739117931168564756083, 5.18990216277976469497697689202, 6.71642186043107513365306655503, 7.34655088173227792151623865003, 8.261056960963711510074226455600, 8.813360443027889967801499031454, 10.16046832407110906429605887247