# Properties

 Degree 2 Conductor $2^{5} \cdot 3^{3} \cdot 7$ Sign $0.258 + 0.965i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.12·5-s − i·7-s + 3.86i·11-s + 5.08i·13-s − 2i·17-s − 7.99·19-s − 7.68·23-s − 3.72·25-s + 4.73·29-s + 4.08i·31-s + 1.12i·35-s − 8.28i·37-s − 8.41i·41-s + 10.0·43-s − 1.93·47-s + ⋯
 L(s)  = 1 − 0.505·5-s − 0.377i·7-s + 1.16i·11-s + 1.41i·13-s − 0.485i·17-s − 1.83·19-s − 1.60·23-s − 0.744·25-s + 0.878·29-s + 0.733i·31-s + 0.190i·35-s − 1.36i·37-s − 1.31i·41-s + 1.52·43-s − 0.282·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6048$$    =    $$2^{5} \cdot 3^{3} \cdot 7$$ $$\varepsilon$$ = $0.258 + 0.965i$ motivic weight = $$1$$ character : $\chi_{6048} (5615, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 6048,\ (\ :1/2),\ 0.258 + 0.965i)$ $L(1)$ $\approx$ $0.8069843359$ $L(\frac12)$ $\approx$ $0.8069843359$ $L(\frac{3}{2})$ 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 + 1.12T + 5T^{2}$$
11 $$1 - 3.86iT - 11T^{2}$$
13 $$1 - 5.08iT - 13T^{2}$$
17 $$1 + 2iT - 17T^{2}$$
19 $$1 + 7.99T + 19T^{2}$$
23 $$1 + 7.68T + 23T^{2}$$
29 $$1 - 4.73T + 29T^{2}$$
31 $$1 - 4.08iT - 31T^{2}$$
37 $$1 + 8.28iT - 37T^{2}$$
41 $$1 + 8.41iT - 41T^{2}$$
43 $$1 - 10.0T + 43T^{2}$$
47 $$1 + 1.93T + 47T^{2}$$
53 $$1 - 7.72T + 53T^{2}$$
59 $$1 + 10.9iT - 59T^{2}$$
61 $$1 - 11.4iT - 61T^{2}$$
67 $$1 + 5.08T + 67T^{2}$$
71 $$1 - 13.6T + 71T^{2}$$
73 $$1 + 2.63T + 73T^{2}$$
79 $$1 + 7.17iT - 79T^{2}$$
83 $$1 + 14.9iT - 83T^{2}$$
89 $$1 + 12.6iT - 89T^{2}$$
97 $$1 - 4.17T + 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

−7.75316538876823892317601122608, −7.28118725990732990531663350183, −6.59776913423956655089710939633, −5.93952445722253791854862366773, −4.78252464010910632731486021199, −4.17347389236523710713219962311, −3.84562625737940863111967052961, −2.29620108616098144077442724052, −1.88837341919346447752905858732, −0.26003176139400920359723001397, 0.806271150537698766524519076454, 2.18248130541583922713924155289, 2.97174111818743424554710859370, 3.85700850885857279857025776868, 4.43674044426087605977334432652, 5.58826379943648067549220115005, 6.03029960208503269942324297943, 6.62653603042762850213953856432, 7.940247504331194053100879028727, 8.127578042994167383708657265591