# 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 + 2i·5-s + 7-s + 5i·13-s + 17-s − 4i·19-s + 5·23-s + 25-s − 9i·29-s + 7·31-s + 2i·35-s + 2i·37-s + 2·41-s − 5i·43-s + 49-s − 9i·53-s + ⋯
 L(s)  = 1 + 0.894i·5-s + 0.377·7-s + 1.38i·13-s + 0.242·17-s − 0.917i·19-s + 1.04·23-s + 0.200·25-s − 1.67i·29-s + 1.25·31-s + 0.338i·35-s + 0.328i·37-s + 0.312·41-s − 0.762i·43-s + 0.142·49-s − 1.23i·53-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 : $\chi_{6048} (3025, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$ $L(1)$ $\approx$ $2.243389270$ $L(\frac12)$ $\approx$ $2.243389270$ $L(\frac{3}{2})$ 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 - T$$
good5 $$1 - 2iT - 5T^{2}$$
11 $$1 - 11T^{2}$$
13 $$1 - 5iT - 13T^{2}$$
17 $$1 - T + 17T^{2}$$
19 $$1 + 4iT - 19T^{2}$$
23 $$1 - 5T + 23T^{2}$$
29 $$1 + 9iT - 29T^{2}$$
31 $$1 - 7T + 31T^{2}$$
37 $$1 - 2iT - 37T^{2}$$
41 $$1 - 2T + 41T^{2}$$
43 $$1 + 5iT - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 9iT - 53T^{2}$$
59 $$1 + iT - 59T^{2}$$
61 $$1 - 6iT - 61T^{2}$$
67 $$1 - 9iT - 67T^{2}$$
71 $$1 - 15T + 71T^{2}$$
73 $$1 + 73T^{2}$$
79 $$1 - 14T + 79T^{2}$$
83 $$1 + 4iT - 83T^{2}$$
89 $$1 + 13T + 89T^{2}$$
97 $$1 + 14T + 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

−8.160121445874212267566844211224, −7.29296081509303595524054842237, −6.75885488534285347256875211226, −6.27963530035108298059024491788, −5.21006649429812336758834395192, −4.52744584124502477203492833870, −3.76819757261829282929161123519, −2.75163516912380957984230910762, −2.16884720685062768093969796126, −0.890840586339115948092554026183, 0.78187672482828431219914192437, 1.46644659443831966827454615355, 2.77789726791239328209640723026, 3.46982435331385974920950612120, 4.53267727581114999926143974025, 5.12834744773377447375616099912, 5.64927597714867348917153816715, 6.55062231685487687322961879923, 7.43015231496483898726799062342, 8.129666030689296445274998829153