# Properties

 Degree 2 Conductor $2^{6} \cdot 3^{2} \cdot 7$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·5-s + 7-s − 4·11-s + 2·13-s + 6·17-s + 4·19-s − 25-s − 2·29-s − 2·35-s − 6·37-s − 2·41-s − 4·43-s + 49-s + 6·53-s + 8·55-s − 12·59-s + 2·61-s − 4·65-s + 4·67-s − 6·73-s − 4·77-s + 16·79-s + 12·83-s − 12·85-s + 14·89-s + 2·91-s − 8·95-s + ⋯
 L(s)  = 1 − 0.894·5-s + 0.377·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 0.338·35-s − 0.986·37-s − 0.312·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s + 1.07·55-s − 1.56·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s − 0.702·73-s − 0.455·77-s + 1.80·79-s + 1.31·83-s − 1.30·85-s + 1.48·89-s + 0.209·91-s − 0.820·95-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4032$$    =    $$2^{6} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{4032} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 4032,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$1.473336744$$ $$L(\frac12)$$ $$\approx$$ $$1.473336744$$ $$L(\frac{3}{2})$$ 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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$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 - 4 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + 2 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 + 6 T + p T^{2}$$
41 $$1 + 2 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 - 2 T + p T^{2}$$
67 $$1 - 4 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 - 16 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 - 14 T + p T^{2}$$
97 $$1 - 18 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

−18.28882051162758, −17.56412888727163, −16.73404337763195, −16.20283349376123, −15.63112999906630, −15.18130576687009, −14.42451995678790, −13.73446231903253, −13.19508063302946, −12.27044893044395, −11.91433295586737, −11.20992491334543, −10.50494844898212, −9.970888617518094, −9.042140676903253, −8.229992153219431, −7.706911835010117, −7.324419367497448, −6.145216521425822, −5.346417456975608, −4.807224522687250, −3.622561990430718, −3.227465062771161, −1.934808584887592, −0.6957792438310278, 0.6957792438310278, 1.934808584887592, 3.227465062771161, 3.622561990430718, 4.807224522687250, 5.346417456975608, 6.145216521425822, 7.324419367497448, 7.706911835010117, 8.229992153219431, 9.042140676903253, 9.970888617518094, 10.50494844898212, 11.20992491334543, 11.91433295586737, 12.27044893044395, 13.19508063302946, 13.73446231903253, 14.42451995678790, 15.18130576687009, 15.63112999906630, 16.20283349376123, 16.73404337763195, 17.56412888727163, 18.28882051162758