# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5 \cdot 7$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 12-s + 2·13-s − 14-s + 15-s + 16-s − 6·17-s − 18-s + 8·19-s + 20-s + 21-s − 24-s + 25-s − 2·26-s + 27-s + 28-s + 6·29-s − 30-s − 4·31-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.218·21-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s − 0.718·31-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$210$$    =    $$2 \cdot 3 \cdot 5 \cdot 7$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{210} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 210,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$1.176663074$$ $$L(\frac12)$$ $$\approx$$ $$1.176663074$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$F_p(T) = 1 - a_p T + p T^2 .$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 + T$$
3 $$1 - T$$
5 $$1 - T$$
7 $$1 - T$$
good11 $$1 + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 - 8 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 + 10 T + p T^{2}$$
41 $$1 + 6 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 + 12 T + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 - 12 T + p T^{2}$$
73 $$1 + 10 T + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 + 10 T + p T^{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

−19.41449765613561, −18.24297464494017, −17.99623847676949, −16.95306223415563, −15.87014077075238, −15.33449676238773, −14.04081082588442, −13.56084994214700, −12.25378438340331, −11.21669057172523, −10.29726946096077, −9.262397060563074, −8.581528352643132, −7.482300394659393, −6.439686434544953, −4.971033039992616, −3.229777727391885, −1.695944514837647, 1.695944514837647, 3.229777727391885, 4.971033039992616, 6.439686434544953, 7.482300394659393, 8.581528352643132, 9.262397060563074, 10.29726946096077, 11.21669057172523, 12.25378438340331, 13.56084994214700, 14.04081082588442, 15.33449676238773, 15.87014077075238, 16.95306223415563, 17.99623847676949, 18.24297464494017, 19.41449765613561