# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7$ Sign $0.318 - 0.947i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 + i)2-s + (−1.21 − 2.74i)3-s − 2i·4-s + (3.32 + 3.73i)5-s + (3.95 + 1.52i)6-s + (−6.61 + 2.29i)7-s + (2 + 2i)8-s + (−6.03 + 6.67i)9-s + (−7.05 − 0.417i)10-s − 10.5i·11-s + (−5.48 + 2.43i)12-s + (14.9 + 14.9i)13-s + (4.31 − 8.90i)14-s + (6.20 − 13.6i)15-s − 4·16-s + (15.4 + 15.4i)17-s + ⋯
 L(s)  = 1 + (−0.5 + 0.5i)2-s + (−0.405 − 0.913i)3-s − 0.5i·4-s + (0.664 + 0.747i)5-s + (0.659 + 0.254i)6-s + (−0.944 + 0.327i)7-s + (0.250 + 0.250i)8-s + (−0.670 + 0.741i)9-s + (−0.705 − 0.0417i)10-s − 0.961i·11-s + (−0.456 + 0.202i)12-s + (1.15 + 1.15i)13-s + (0.308 − 0.636i)14-s + (0.413 − 0.910i)15-s − 0.250·16-s + (0.911 + 0.911i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $0.318 - 0.947i$ motivic weight = $$2$$ character : $\chi_{210} (167, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 210,\ (\ :1),\ 0.318 - 0.947i)$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.764369 + 0.549451i$$ $$L(\frac12)$$ $$\approx$$ $$0.764369 + 0.549451i$$ $$L(2)$$ 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 - i)T$$
3 $$1 + (1.21 + 2.74i)T$$
5 $$1 + (-3.32 - 3.73i)T$$
7 $$1 + (6.61 - 2.29i)T$$
good11 $$1 + 10.5iT - 121T^{2}$$
13 $$1 + (-14.9 - 14.9i)T + 169iT^{2}$$
17 $$1 + (-15.4 - 15.4i)T + 289iT^{2}$$
19 $$1 + 17.3T + 361T^{2}$$
23 $$1 + (-23.1 - 23.1i)T + 529iT^{2}$$
29 $$1 - 23.7T + 841T^{2}$$
31 $$1 - 33.1iT - 961T^{2}$$
37 $$1 + (17.6 + 17.6i)T + 1.36e3iT^{2}$$
41 $$1 + 11.8T + 1.68e3T^{2}$$
43 $$1 + (22.8 - 22.8i)T - 1.84e3iT^{2}$$
47 $$1 + (-12.6 - 12.6i)T + 2.20e3iT^{2}$$
53 $$1 + (-15.3 - 15.3i)T + 2.80e3iT^{2}$$
59 $$1 - 31.0iT - 3.48e3T^{2}$$
61 $$1 + 48.6iT - 3.72e3T^{2}$$
67 $$1 + (77.5 + 77.5i)T + 4.48e3iT^{2}$$
71 $$1 + 60.7iT - 5.04e3T^{2}$$
73 $$1 + (-3.52 - 3.52i)T + 5.32e3iT^{2}$$
79 $$1 - 99.4iT - 6.24e3T^{2}$$
83 $$1 + (-16.9 + 16.9i)T - 6.88e3iT^{2}$$
89 $$1 + 17.6iT - 7.92e3T^{2}$$
97 $$1 + (34.7 - 34.7i)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

−12.38688357076596517720553745156, −11.15729311691778755927410702930, −10.50361513602889776328515771071, −9.190923926367424097201891675731, −8.353787601306633765718464388133, −6.92758982250962424875778811112, −6.32477967342986768290092643886, −5.65302001657038641928334117733, −3.22728474617021400726303282380, −1.51484764938383600636000144372, 0.70474089958181528098046882597, 2.94556192807030361833316852945, 4.30288271670881539228606396571, 5.52087569782235359284714978957, 6.70077467816957727604633644891, 8.380265879603198548046084664969, 9.253776233425263064813582004472, 10.14371326425731602887942261600, 10.52097191494083836875333483543, 11.93466999138225339543732260064