Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.833 - 0.552i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)3-s + 1.61·7-s + (−0.951 − 1.30i)13-s + (0.587 + 0.190i)19-s + (0.500 + 1.53i)21-s + (−0.809 − 0.587i)23-s + (0.809 + 0.587i)27-s + (0.190 + 0.587i)29-s + (0.951 + 0.309i)31-s + (0.587 + 0.809i)37-s + (0.951 − 1.30i)39-s + 0.618·43-s + (−0.5 − 1.53i)47-s + 1.61·49-s + (−0.951 + 0.309i)53-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)3-s + 1.61·7-s + (−0.951 − 1.30i)13-s + (0.587 + 0.190i)19-s + (0.500 + 1.53i)21-s + (−0.809 − 0.587i)23-s + (0.809 + 0.587i)27-s + (0.190 + 0.587i)29-s + (0.951 + 0.309i)31-s + (0.587 + 0.809i)37-s + (0.951 − 1.30i)39-s + 0.618·43-s + (−0.5 − 1.53i)47-s + 1.61·49-s + (−0.951 + 0.309i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.833 - 0.552i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (799, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ 0.833 - 0.552i)$
$L(\frac{1}{2})$  $\approx$  $1.783315836$
$L(\frac12)$  $\approx$  $1.783315836$
$L(1)$   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 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 - 1.61T + T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
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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

−8.513923029490471018549814470936, −8.115337255113439146147754447597, −7.49299795971589265371172666704, −6.48851456591399040887447506492, −5.32060289370328324404104145303, −4.94359282659929912133959401486, −4.27926016418992816742135405224, −3.31378896328819707873125047547, −2.43685760503564186696678873216, −1.21353372159341419248320091114, 1.29783510108590529105028596367, 1.97322287210696176740920862652, 2.71172869604517473426881947558, 4.31360695967527327994523736947, 4.58880446137308658945649355976, 5.61881503924697154934373624566, 6.52503560481823514820035519029, 7.32618924120938795641078568337, 7.80915052021029706615614760468, 8.221307232432302404824957489271

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