# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 11 \cdot 17^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + 4-s − 2·7-s + 8-s − 11-s − 4·13-s − 2·14-s + 16-s − 4·19-s − 22-s + 6·23-s − 5·25-s − 4·26-s − 2·28-s + 6·29-s − 8·31-s + 32-s + 10·37-s − 4·38-s + 6·41-s + 8·43-s − 44-s + 6·46-s + 6·47-s − 3·49-s − 5·50-s − 4·52-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 0.301·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.917·19-s − 0.213·22-s + 1.25·23-s − 25-s − 0.784·26-s − 0.377·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 1.64·37-s − 0.648·38-s + 0.937·41-s + 1.21·43-s − 0.150·44-s + 0.884·46-s + 0.875·47-s − 3/7·49-s − 0.707·50-s − 0.554·52-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 57222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 57222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$57222$$    =    $$2 \cdot 3^{2} \cdot 11 \cdot 17^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{57222} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 57222,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;11,\;17\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - T$$
3 $$1$$
11 $$1 + T$$
17 $$1$$
good5 $$1 + p T^{2}$$
7 $$1 + 2 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 - 6 T + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 + 8 T + p T^{2}$$
37 $$1 - 10 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 - 8 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + 8 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 - 6 T + p T^{2}$$
73 $$1 + 2 T + p T^{2}$$
79 $$1 + 14 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 + 14 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

−14.63917718491213, −14.14118717436648, −13.53509834236016, −13.01185763411863, −12.64584504154000, −12.31510876453235, −11.68012595404732, −10.93322720466951, −10.76367957690681, −10.01964282591604, −9.431420879070070, −9.161810285238644, −8.283806951536278, −7.671802963374669, −7.223132463467104, −6.720066100617865, −5.960520268394544, −5.741323927958606, −4.865171997141630, −4.424406129468090, −3.862361409858416, −3.013036416644023, −2.620974773614891, −2.007285676274223, −0.9262751546285971, 0, 0.9262751546285971, 2.007285676274223, 2.620974773614891, 3.013036416644023, 3.862361409858416, 4.424406129468090, 4.865171997141630, 5.741323927958606, 5.960520268394544, 6.720066100617865, 7.223132463467104, 7.671802963374669, 8.283806951536278, 9.161810285238644, 9.431420879070070, 10.01964282591604, 10.76367957690681, 10.93322720466951, 11.68012595404732, 12.31510876453235, 12.64584504154000, 13.01185763411863, 13.53509834236016, 14.14118717436648, 14.63917718491213