# Properties

 Degree $2$ Conductor $1050$ Sign $-0.0899 - 0.995i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − i·2-s + (−0.721 + 1.57i)3-s − 4-s + (1.57 + 0.721i)6-s + (−1.31 − 2.29i)7-s + i·8-s + (−1.95 − 2.27i)9-s + 3.91i·11-s + (0.721 − 1.57i)12-s − 4.99i·13-s + (−2.29 + 1.31i)14-s + 16-s + 3.54·17-s + (−2.27 + 1.95i)18-s + 3.14i·19-s + ⋯
 L(s)  = 1 − 0.707i·2-s + (−0.416 + 0.909i)3-s − 0.5·4-s + (0.642 + 0.294i)6-s + (−0.496 − 0.867i)7-s + 0.353i·8-s + (−0.652 − 0.757i)9-s + 1.18i·11-s + (0.208 − 0.454i)12-s − 1.38i·13-s + (−0.613 + 0.351i)14-s + 0.250·16-s + 0.860·17-s + (−0.535 + 0.461i)18-s + 0.722i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0899 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0899 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1050$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 7$$ Sign: $-0.0899 - 0.995i$ Motivic weight: $$1$$ Character: $\chi_{1050} (251, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1050,\ (\ :1/2),\ -0.0899 - 0.995i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6393145263$$ $$L(\frac12)$$ $$\approx$$ $$0.6393145263$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 + (0.721 - 1.57i)T$$
5 $$1$$
7 $$1 + (1.31 + 2.29i)T$$
good11 $$1 - 3.91iT - 11T^{2}$$
13 $$1 + 4.99iT - 13T^{2}$$
17 $$1 - 3.54T + 17T^{2}$$
19 $$1 - 3.14iT - 19T^{2}$$
23 $$1 - 7.54iT - 23T^{2}$$
29 $$1 - 7.54iT - 29T^{2}$$
31 $$1 - 4.19iT - 31T^{2}$$
37 $$1 + 10.4T + 37T^{2}$$
41 $$1 + 9.32T + 41T^{2}$$
43 $$1 + 2.91T + 43T^{2}$$
47 $$1 + 8.00T + 47T^{2}$$
53 $$1 + 0.288iT - 53T^{2}$$
59 $$1 - 5.89T + 59T^{2}$$
61 $$1 - 2.48iT - 61T^{2}$$
67 $$1 - 0.545T + 67T^{2}$$
71 $$1 - 5.37iT - 71T^{2}$$
73 $$1 - 4.85iT - 73T^{2}$$
79 $$1 - 0.742T + 79T^{2}$$
83 $$1 - 4.45T + 83T^{2}$$
89 $$1 - 12.3T + 89T^{2}$$
97 $$1 - 0.524iT - 97T^{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

−10.12910443499150900951828097992, −9.829417270946106511454061581481, −8.732132119220194522333556258798, −7.69035950343014821531485196653, −6.77789567177742632584997640483, −5.42755859379224433806433804616, −4.96903305048187364115507317282, −3.56137376964583380254074398501, −3.35971592558128000491386456495, −1.38924976510596922884669902560, 0.31753769077783261438014609348, 2.06829389242040314291249005723, 3.30930975316153323217875783456, 4.77361322643616952773953042359, 5.68415124167787086399133003714, 6.43416939953119687213207811054, 6.86164223992845623724128355250, 8.124173830747916729141474796253, 8.595250201391146561405826523941, 9.436532464307206658768004995540