# Properties

 Degree $2$ Conductor $210$ Sign $0.487 + 0.872i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − i·2-s + (−1.61 + 0.618i)3-s − 4-s + 5-s + (0.618 + 1.61i)6-s + (2.61 − 0.381i)7-s + i·8-s + (2.23 − 2.00i)9-s − i·10-s − 4.47i·11-s + (1.61 − 0.618i)12-s + 1.23i·13-s + (−0.381 − 2.61i)14-s + (−1.61 + 0.618i)15-s + 16-s + 5.23·17-s + ⋯
 L(s)  = 1 − 0.707i·2-s + (−0.934 + 0.356i)3-s − 0.5·4-s + 0.447·5-s + (0.252 + 0.660i)6-s + (0.989 − 0.144i)7-s + 0.353i·8-s + (0.745 − 0.666i)9-s − 0.316i·10-s − 1.34i·11-s + (0.467 − 0.178i)12-s + 0.342i·13-s + (−0.102 − 0.699i)14-s + (−0.417 + 0.159i)15-s + 0.250·16-s + 1.26·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ Sign: $0.487 + 0.872i$ Motivic weight: $$1$$ Character: $\chi_{210} (41, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 210,\ (\ :1/2),\ 0.487 + 0.872i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.885203 - 0.519284i$$ $$L(\frac12)$$ $$\approx$$ $$0.885203 - 0.519284i$$ $$L(\frac{3}{2})$$ 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 + (1.61 - 0.618i)T$$
5 $$1 - T$$
7 $$1 + (-2.61 + 0.381i)T$$
good11 $$1 + 4.47iT - 11T^{2}$$
13 $$1 - 1.23iT - 13T^{2}$$
17 $$1 - 5.23T + 17T^{2}$$
19 $$1 + 8.47iT - 19T^{2}$$
23 $$1 - 4iT - 23T^{2}$$
29 $$1 - 7.70iT - 29T^{2}$$
31 $$1 - 2.76iT - 31T^{2}$$
37 $$1 - 0.763T + 37T^{2}$$
41 $$1 + 2.47T + 41T^{2}$$
43 $$1 + 4.94T + 43T^{2}$$
47 $$1 + 6.47T + 47T^{2}$$
53 $$1 + 0.472iT - 53T^{2}$$
59 $$1 - 4.47T + 59T^{2}$$
61 $$1 + 7.23iT - 61T^{2}$$
67 $$1 + 12T + 67T^{2}$$
71 $$1 - 7.23iT - 71T^{2}$$
73 $$1 - 11.2iT - 73T^{2}$$
79 $$1 + 8.94T + 79T^{2}$$
83 $$1 - 14.6T + 83T^{2}$$
89 $$1 + 5.52T + 89T^{2}$$
97 $$1 + 0.763iT - 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

−11.83055703838193839995996042041, −11.24251325779676715312717541218, −10.57881849207187713242554907023, −9.478362294525609883146800929672, −8.494393243429295623907305615873, −6.98616253069062895879214867105, −5.56250779248751694961822887916, −4.85894466694509251136112359601, −3.33016707163840279440937490097, −1.21527653637586685339144657534, 1.70144517042571448310252289478, 4.37719302623411572614881697721, 5.39378971821176985731350793601, 6.21026672455125969005904546329, 7.52206610716490829259471236966, 8.111410762312666473310373948238, 9.842717578474260972633046809822, 10.37728005729049153696305905483, 11.87415649155267587468497862383, 12.38597062050940731012397942374