# Properties

 Degree 2 Conductor $2^{5} \cdot 3^{3} \cdot 7$ Sign $0.707 - 0.707i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.66i·5-s − i·7-s + 0.164·11-s + 3.56·13-s + 3.41i·17-s + 3.89i·19-s + 5.29·23-s − 2.09·25-s − 7.84i·29-s − 8.72i·31-s + 2.66·35-s + 7.82·37-s + 1.62i·41-s − 11.2i·43-s − 0.585·47-s + ⋯
 L(s)  = 1 + 1.19i·5-s − 0.377i·7-s + 0.0496·11-s + 0.987·13-s + 0.828i·17-s + 0.894i·19-s + 1.10·23-s − 0.419·25-s − 1.45i·29-s − 1.56i·31-s + 0.450·35-s + 1.28·37-s + 0.254i·41-s − 1.71i·43-s − 0.0854·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6048$$    =    $$2^{5} \cdot 3^{3} \cdot 7$$ $$\varepsilon$$ = $0.707 - 0.707i$ motivic weight = $$1$$ character : $\chi_{6048} (2591, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$ $L(1)$ $\approx$ $2.211227443$ $L(\frac12)$ $\approx$ $2.211227443$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 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 + iT$$
good5 $$1 - 2.66iT - 5T^{2}$$
11 $$1 - 0.164T + 11T^{2}$$
13 $$1 - 3.56T + 13T^{2}$$
17 $$1 - 3.41iT - 17T^{2}$$
19 $$1 - 3.89iT - 19T^{2}$$
23 $$1 - 5.29T + 23T^{2}$$
29 $$1 + 7.84iT - 29T^{2}$$
31 $$1 + 8.72iT - 31T^{2}$$
37 $$1 - 7.82T + 37T^{2}$$
41 $$1 - 1.62iT - 41T^{2}$$
43 $$1 + 11.2iT - 43T^{2}$$
47 $$1 + 0.585T + 47T^{2}$$
53 $$1 - 14.2iT - 53T^{2}$$
59 $$1 - 6.98T + 59T^{2}$$
61 $$1 - 4.92T + 61T^{2}$$
67 $$1 + 6.83iT - 67T^{2}$$
71 $$1 + 10.2T + 71T^{2}$$
73 $$1 - 5.48T + 73T^{2}$$
79 $$1 - 9.09iT - 79T^{2}$$
83 $$1 - 1.95T + 83T^{2}$$
89 $$1 + 6.05iT - 89T^{2}$$
97 $$1 - 4.72T + 97T^{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

−7.941012063077968096605612787244, −7.55851052193242477671375439531, −6.62164424443615967223879181877, −6.17220848811309329329673863686, −5.53573730471905349083846028835, −4.20145220445890667517662287387, −3.82914193929649341956614969201, −2.91376992838494018089215133427, −2.08198856930287828568411354558, −0.886379589141649775447588503392, 0.77759432400983154397563839760, 1.47123304723352687171768168702, 2.75621253919212765250398720042, 3.46970426901773895674344867857, 4.65528017714903491502655454887, 4.97243278526274039741619143602, 5.69210259520315760941970806662, 6.66392647893696558722244812999, 7.17456500904320416310208782108, 8.274545809774457552365757268860