# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7$ Sign $-0.772 - 0.634i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.36 + 0.366i)2-s + (0.145 + 2.99i)3-s + (1.73 + i)4-s + (−4.92 + 0.860i)5-s + (−0.898 + 4.14i)6-s + (1.68 + 6.79i)7-s + (1.99 + 2i)8-s + (−8.95 + 0.871i)9-s + (−7.04 − 0.627i)10-s + (−13.7 − 7.94i)11-s + (−2.74 + 5.33i)12-s + (5.32 + 5.32i)13-s + (−0.185 + 9.89i)14-s + (−3.29 − 14.6i)15-s + (1.99 + 3.46i)16-s + (16.9 − 4.55i)17-s + ⋯
 L(s)  = 1 + (0.683 + 0.183i)2-s + (0.0484 + 0.998i)3-s + (0.433 + 0.250i)4-s + (−0.985 + 0.172i)5-s + (−0.149 + 0.691i)6-s + (0.240 + 0.970i)7-s + (0.249 + 0.250i)8-s + (−0.995 + 0.0968i)9-s + (−0.704 − 0.0627i)10-s + (−1.25 − 0.722i)11-s + (−0.228 + 0.444i)12-s + (0.409 + 0.409i)13-s + (−0.0132 + 0.706i)14-s + (−0.219 − 0.975i)15-s + (0.124 + 0.216i)16-s + (0.999 − 0.267i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $-0.772 - 0.634i$ motivic weight = $$2$$ character : $\chi_{210} (17, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 210,\ (\ :1),\ -0.772 - 0.634i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.561882 + 1.56903i$$ $$L(\frac12)$$ $$\approx$$ $$0.561882 + 1.56903i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-1.36 - 0.366i)T$$
3 $$1 + (-0.145 - 2.99i)T$$
5 $$1 + (4.92 - 0.860i)T$$
7 $$1 + (-1.68 - 6.79i)T$$
good11 $$1 + (13.7 + 7.94i)T + (60.5 + 104. i)T^{2}$$
13 $$1 + (-5.32 - 5.32i)T + 169iT^{2}$$
17 $$1 + (-16.9 + 4.55i)T + (250. - 144.5i)T^{2}$$
19 $$1 + (-5.62 - 9.74i)T + (-180.5 + 312. i)T^{2}$$
23 $$1 + (10.4 - 39.1i)T + (-458. - 264.5i)T^{2}$$
29 $$1 + 22.8T + 841T^{2}$$
31 $$1 + (-36.0 - 20.8i)T + (480.5 + 832. i)T^{2}$$
37 $$1 + (-7.72 + 28.8i)T + (-1.18e3 - 684.5i)T^{2}$$
41 $$1 - 55.0T + 1.68e3T^{2}$$
43 $$1 + (14.2 - 14.2i)T - 1.84e3iT^{2}$$
47 $$1 + (6.77 - 25.2i)T + (-1.91e3 - 1.10e3i)T^{2}$$
53 $$1 + (-59.0 + 15.8i)T + (2.43e3 - 1.40e3i)T^{2}$$
59 $$1 + (-10.7 - 6.18i)T + (1.74e3 + 3.01e3i)T^{2}$$
61 $$1 + (-14.5 + 8.40i)T + (1.86e3 - 3.22e3i)T^{2}$$
67 $$1 + (-103. + 27.8i)T + (3.88e3 - 2.24e3i)T^{2}$$
71 $$1 + 34.7iT - 5.04e3T^{2}$$
73 $$1 + (56.8 - 15.2i)T + (4.61e3 - 2.66e3i)T^{2}$$
79 $$1 + (-40.5 + 23.4i)T + (3.12e3 - 5.40e3i)T^{2}$$
83 $$1 + (-75.6 + 75.6i)T - 6.88e3iT^{2}$$
89 $$1 + (116. - 67.1i)T + (3.96e3 - 6.85e3i)T^{2}$$
97 $$1 + (75.7 - 75.7i)T - 9.40e3iT^{2}$$
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\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

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

−12.31898545809875219837155726441, −11.53314131832070149260518009017, −10.89092423433958233912549364926, −9.620283498419673905073278369577, −8.359415369466242936153390447686, −7.68937584309819821241499640795, −5.84591625970270956563864949156, −5.17605730920267577432342578317, −3.80551330695266203296631105707, −2.88474792796821257696299503368, 0.75806687225865809819225823030, 2.67677466503763542067090616273, 4.09308265974775899135853671338, 5.30664684243066333640747015734, 6.74583511811000700559206359097, 7.69347417907558282391389760473, 8.221455122924366391308168524391, 10.17291967748177576916357711263, 11.03854308295443314009839315599, 11.99322519800398757840998305926