# Properties

 Degree 2 Conductor $2^{4} \cdot 3^{2} \cdot 7$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

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## Dirichlet series

 L(s)  = 1 − 2·5-s + 7-s − 4·11-s + 2·13-s + 6·17-s − 8·19-s − 25-s − 6·29-s − 8·31-s − 2·35-s − 2·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 6·53-s + 8·55-s − 6·61-s − 4·65-s + 4·67-s − 8·71-s + 10·73-s − 4·77-s − 16·79-s + 8·83-s − 12·85-s + 6·89-s + ⋯
 L(s)  = 1 − 0.894·5-s + 0.377·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s − 0.768·61-s − 0.496·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.455·77-s − 1.80·79-s + 0.878·83-s − 1.30·85-s + 0.635·89-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1008$$    =    $$2^{4} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{1008} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 1008,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;7\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1$$
7 $$1 - T$$
good5 $$1 + 2 T + p T^{2}$$
11 $$1 + 4 T + 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 + 8 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 + 2 T + p T^{2}$$
43 $$1 - 4 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + 6 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 - 10 T + p T^{2}$$
79 $$1 + 16 T + p T^{2}$$
83 $$1 - 8 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 + 6 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

−19.81091954122851, −18.84302431198750, −18.76537182946652, −17.80542961159119, −16.92676191659814, −16.30219183151636, −15.59822957511555, −14.91621165711342, −14.36962942814042, −13.22122852644190, −12.72249388729552, −11.92211761661144, −10.98784923157337, −10.63145032630949, −9.559817840156688, −8.471445676889714, −7.930145034161771, −7.275542518020635, −6.021997044143567, −5.194033396068595, −4.123066989008585, −3.275995389802918, −1.865371178942683, 0, 1.865371178942683, 3.275995389802918, 4.123066989008585, 5.194033396068595, 6.021997044143567, 7.275542518020635, 7.930145034161771, 8.471445676889714, 9.559817840156688, 10.63145032630949, 10.98784923157337, 11.92211761661144, 12.72249388729552, 13.22122852644190, 14.36962942814042, 14.91621165711342, 15.59822957511555, 16.30219183151636, 16.92676191659814, 17.80542961159119, 18.76537182946652, 18.84302431198750, 19.81091954122851