Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $-0.715 - 0.698i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $-0.715 - 0.698i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (2399, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ -0.715 - 0.698i)$
$L(\frac{1}{2})$  $\approx$  $0.001349061221$
$L(\frac12)$  $\approx$  $0.001349061221$
$L(1)$   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

−8.245715467393627445374602541742, −7.01230645882854777620421640941, −6.70312511570715223156163778469, −6.27096509829528386896867986702, −5.05999693966374599208256923731, −4.70186060927938040702545539008, −3.36669611713169556130848060550, −2.60929555515067137449739993231, −1.32194223689711514608347402316, −0.000846644362155833575642995893, 1.78685198367493889744002326325, 2.93147685622055406538461647097, 3.84983018069211720660885462636, 4.67903373156426047076434956911, 5.31729060359090405236493951837, 6.18437599687721968710701797649, 6.57005145961567404977108999231, 7.77482580826659222579334553655, 8.203017985838717898609751914291, 9.264185172255693344713925188055

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