# Properties

 Degree $2$ Conductor $1400$ Sign $-0.742 + 0.669i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.866 + 0.5i)3-s + (−1.73 + 2i)7-s + (−1 − 1.73i)9-s + (−1.5 + 2.59i)11-s − 6i·13-s + (−4.33 − 2.5i)17-s + (0.5 + 0.866i)19-s + (−2.5 + 0.866i)21-s + (−6.06 + 3.5i)23-s − 5i·27-s − 2·29-s + (2.5 − 4.33i)31-s + (−2.59 + 1.5i)33-s + (−2.59 + 1.5i)37-s + (3 − 5.19i)39-s + ⋯
 L(s)  = 1 + (0.499 + 0.288i)3-s + (−0.654 + 0.755i)7-s + (−0.333 − 0.577i)9-s + (−0.452 + 0.783i)11-s − 1.66i·13-s + (−1.05 − 0.606i)17-s + (0.114 + 0.198i)19-s + (−0.545 + 0.188i)21-s + (−1.26 + 0.729i)23-s − 0.962i·27-s − 0.371·29-s + (0.449 − 0.777i)31-s + (−0.452 + 0.261i)33-s + (−0.427 + 0.246i)37-s + (0.480 − 0.832i)39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1400$$    =    $$2^{3} \cdot 5^{2} \cdot 7$$ Sign: $-0.742 + 0.669i$ Motivic weight: $$1$$ Character: $\chi_{1400} (249, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1400,\ (\ :1/2),\ -0.742 + 0.669i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.3907040812$$ $$L(\frac12)$$ $$\approx$$ $$0.3907040812$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
7 $$1 + (1.73 - 2i)T$$
good3 $$1 + (-0.866 - 0.5i)T + (1.5 + 2.59i)T^{2}$$
11 $$1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + 6iT - 13T^{2}$$
17 $$1 + (4.33 + 2.5i)T + (8.5 + 14.7i)T^{2}$$
19 $$1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (6.06 - 3.5i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + 2T + 29T^{2}$$
31 $$1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (2.59 - 1.5i)T + (18.5 - 32.0i)T^{2}$$
41 $$1 + 2T + 41T^{2}$$
43 $$1 + 4iT - 43T^{2}$$
47 $$1 + (4.33 - 2.5i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (-0.866 - 0.5i)T + (26.5 + 45.8i)T^{2}$$
59 $$1 + (-7.5 + 12.9i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (7.79 + 4.5i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (6.06 + 3.5i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 12iT - 83T^{2}$$
89 $$1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 - 2iT - 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

−9.449341312307234054673209981501, −8.428129383748826318371661041744, −7.86128129450520321142914138477, −6.76874181712498439937589566431, −5.87592895374555893902510125540, −5.15933199347355835271774348098, −3.90822166245208246362551537234, −3.02976048155125047227069162442, −2.23570270255096852800178402951, −0.13475906297086189953520179102, 1.74988872079546523244696139985, 2.75169211778699393149820739983, 3.85941950528837757437702710607, 4.64353473214225688645223870650, 5.93926047337008311897925882340, 6.70057596776158568517647903846, 7.39801059638939267925580776571, 8.456818830885457868594111081994, 8.809589872147708968750312303238, 9.882870927344511970653481343681