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

Origins

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Normalization:  

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

Graph of the $Z$-function along the critical line