Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.994 - 0.105i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 + 0.809i)9-s + (1.76 + 0.278i)13-s + (−0.896 − 1.76i)17-s + (1.11 − 0.363i)29-s + (0.0489 − 0.309i)37-s + (−1.53 + 1.11i)41-s + i·49-s + (0.412 − 0.809i)53-s + (1.53 + 1.11i)61-s + (0.142 + 0.896i)73-s + (−0.309 + 0.951i)81-s + (0.690 − 0.951i)89-s + (−0.278 − 0.142i)97-s + 1.61·101-s + (−0.363 − 0.5i)109-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)9-s + (1.76 + 0.278i)13-s + (−0.896 − 1.76i)17-s + (1.11 − 0.363i)29-s + (0.0489 − 0.309i)37-s + (−1.53 + 1.11i)41-s + i·49-s + (0.412 − 0.809i)53-s + (1.53 + 1.11i)61-s + (0.142 + 0.896i)73-s + (−0.309 + 0.951i)81-s + (0.690 − 0.951i)89-s + (−0.278 − 0.142i)97-s + 1.61·101-s + (−0.363 − 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.994 - 0.105i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (1793, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ 0.994 - 0.105i)$
$L(\frac{1}{2})$  $\approx$  $1.465384419$
$L(\frac12)$  $\approx$  $1.465384419$
$L(1)$   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 \( 1 + (-0.587 - 0.809i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.951 + 0.309i)T^{2} \)
29 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.0489 + 0.309i)T + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (-0.412 + 0.809i)T + (-0.587 - 0.809i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.587 + 0.809i)T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.587 - 0.809i)T^{2} \)
89 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2} \)
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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

−8.572826510963427494698232511226, −8.024996762893183696577410326444, −7.02519498708550208598214563735, −6.61958689752670733596629044534, −5.64562952951277556262959605093, −4.79155203502633540688866692971, −4.20064040665824189601359278718, −3.15585883393548520857470050963, −2.21647296787341595526501109389, −1.11519974014613452199786838663, 1.09943522073885087795436848309, 2.02921698656194227111275542751, 3.52244590849793936171477582073, 3.78078493873032308245740174740, 4.78554773091469377040526984307, 5.85173660139586633968019129454, 6.45012110743747835061881979040, 6.91291594940670005380097965151, 8.140434587313513454323523124661, 8.566934501482305945741206327626

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