# Properties

 Degree 2 Conductor $2^{5} \cdot 5^{3}$ Sign $0.327 + 0.944i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.809 − 0.587i)3-s − 0.618·7-s + (0.587 − 0.190i)13-s + (0.951 + 1.30i)19-s + (0.500 + 0.363i)21-s + (0.309 − 0.951i)23-s + (−0.309 + 0.951i)27-s + (1.30 + 0.951i)29-s + (−0.587 − 0.809i)31-s + (0.951 − 0.309i)37-s + (−0.587 − 0.190i)39-s − 1.61·43-s + (−0.5 − 0.363i)47-s − 0.618·49-s + (0.587 − 0.809i)53-s + ⋯
 L(s)  = 1 + (−0.809 − 0.587i)3-s − 0.618·7-s + (0.587 − 0.190i)13-s + (0.951 + 1.30i)19-s + (0.500 + 0.363i)21-s + (0.309 − 0.951i)23-s + (−0.309 + 0.951i)27-s + (1.30 + 0.951i)29-s + (−0.587 − 0.809i)31-s + (0.951 − 0.309i)37-s + (−0.587 − 0.190i)39-s − 1.61·43-s + (−0.5 − 0.363i)47-s − 0.618·49-s + (0.587 − 0.809i)53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4000$$    =    $$2^{5} \cdot 5^{3}$$ $$\varepsilon$$ = $0.327 + 0.944i$ motivic weight = $$0$$ character : $\chi_{4000} (2399, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4000,\ (\ :0),\ 0.327 + 0.944i)$ $L(\frac{1}{2})$ $\approx$ $0.8635128577$ $L(\frac12)$ $\approx$ $0.8635128577$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2}$$
7 $$1 + 0.618T + T^{2}$$
11 $$1 + (0.809 + 0.587i)T^{2}$$
13 $$1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2}$$
17 $$1 + (-0.309 + 0.951i)T^{2}$$
19 $$1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2}$$
23 $$1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2}$$
29 $$1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2}$$
31 $$1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2}$$
37 $$1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2}$$
41 $$1 + (-0.809 + 0.587i)T^{2}$$
43 $$1 + 1.61T + T^{2}$$
47 $$1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}$$
53 $$1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2}$$
59 $$1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2}$$
61 $$1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2}$$
67 $$1 + (0.309 - 0.951i)T^{2}$$
71 $$1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2}$$
73 $$1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2}$$
79 $$1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2}$$
83 $$1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}$$
89 $$1 + (-0.809 - 0.587i)T^{2}$$
97 $$1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)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

−8.379219112727580198434890957639, −7.71380012491779061842935042825, −6.68560684437288558207663474071, −6.45872621711591649898604355649, −5.62241510746306114841708313192, −4.96299830368133065027202713188, −3.72543614723940592243025582855, −3.12934380897232612678322026610, −1.77112715521194536998217613180, −0.67883757351304709131047255957, 1.04167940902770683982657533805, 2.56797105004540820397226871596, 3.40705796161011591993973174261, 4.36391136422109662963592604527, 5.08421907892918753901910355842, 5.70673748637457216814895324191, 6.52623023642738938175352059933, 7.11865791980504650182398558762, 8.101744690611548006633112364858, 8.840289945623616185995603484929