Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.790 + 0.612i$
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 + (0.896 − 0.142i)13-s + (0.142 − 0.278i)17-s + (−1.11 − 0.363i)29-s + (−0.309 − 1.95i)37-s + (1.53 + 1.11i)41-s i·49-s + (0.809 + 1.58i)53-s + (−1.53 + 1.11i)61-s + (0.278 − 1.76i)73-s + (−0.309 − 0.951i)81-s + (−0.690 − 0.951i)89-s + (1.76 − 0.896i)97-s + 1.61·101-s + (−0.363 + 0.5i)109-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)9-s + (0.896 − 0.142i)13-s + (0.142 − 0.278i)17-s + (−1.11 − 0.363i)29-s + (−0.309 − 1.95i)37-s + (1.53 + 1.11i)41-s i·49-s + (0.809 + 1.58i)53-s + (−1.53 + 1.11i)61-s + (0.278 − 1.76i)73-s + (−0.309 − 0.951i)81-s + (−0.690 − 0.951i)89-s + (1.76 − 0.896i)97-s + 1.61·101-s + (−0.363 + 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.790 + 0.612i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (2657, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ 0.790 + 0.612i)$
$L(\frac{1}{2})$  $\approx$  $1.396995945$
$L(\frac12)$  $\approx$  $1.396995945$
$L(1)$   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 \( 1 + (-0.896 + 0.142i)T + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \)
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 \( 1 + (0.309 + 1.95i)T + (-0.951 + 0.309i)T^{2} \)
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 \( 1 + (-0.809 - 1.58i)T + (-0.587 + 0.809i)T^{2} \)
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 \( 1 + (-0.278 + 1.76i)T + (-0.951 - 0.309i)T^{2} \)
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 \( 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2} \)
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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

−8.778567880969994457901475340473, −7.54853444452210774567697077109, −7.33213064179044085789625021096, −6.09434773798284084682200634270, −5.89740148548473940367181299954, −4.66020254663400576732024428217, −3.92421522720387031243609628335, −3.22599090325069779975953804756, −2.03401547895775351476235100903, −0.898831286554477018862416071550, 1.30743665669055765764464300510, 2.20348215493208934222310816228, 3.36066973990904861418209705415, 4.12947505690685488613356921454, 4.95707737001560491641149482377, 5.73192937273667745825583111487, 6.52257046765086949200433133824, 7.30129800685927133137385957413, 7.971341292194898766453443013427, 8.630649948431706275251795802346

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