Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.0941 - 0.995i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 − 0.309i)9-s + (−1.76 + 0.896i)13-s + (0.896 − 0.142i)17-s + (1.11 + 1.53i)29-s + (0.809 + 1.58i)37-s + (−0.363 + 1.11i)41-s + i·49-s + (−0.309 − 0.0489i)53-s + (0.363 + 1.11i)61-s + (−0.142 + 0.278i)73-s + (0.809 + 0.587i)81-s + (−1.80 + 0.587i)89-s + (0.278 − 1.76i)97-s − 0.618·101-s + (−1.53 − 0.5i)109-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)9-s + (−1.76 + 0.896i)13-s + (0.896 − 0.142i)17-s + (1.11 + 1.53i)29-s + (0.809 + 1.58i)37-s + (−0.363 + 1.11i)41-s + i·49-s + (−0.309 − 0.0489i)53-s + (0.363 + 1.11i)61-s + (−0.142 + 0.278i)73-s + (0.809 + 0.587i)81-s + (−1.80 + 0.587i)89-s + (0.278 − 1.76i)97-s − 0.618·101-s + (−1.53 − 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0941 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0941 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.0941 - 0.995i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (3393, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ 0.0941 - 0.995i)$
$L(\frac{1}{2})$  $\approx$  $0.8524204791$
$L(\frac12)$  $\approx$  $0.8524204791$
$L(1)$   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 \( 1 + (0.951 + 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.896 + 0.142i)T + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 1.58i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.309 + 0.0489i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.278 + 1.76i)T + (-0.951 - 0.309i)T^{2} \)
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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

−8.727632290513133356239263873421, −8.146103572199910578698194573441, −7.26112129689592523098074476606, −6.68804698416045192976207335732, −5.82775090448766577181918602843, −4.99205768008191945147831302792, −4.43311906660818571780168113229, −3.12291237010837719250098840176, −2.67601069272635916747390973873, −1.33089168249077969050786231808, 0.48071580671073338346511777072, 2.23057068338761059646618022487, 2.79372034512748355614855077828, 3.79701417062546298362384664642, 4.87006859435132667612283396699, 5.44108096429349854330290763003, 6.10486807889811791485659495572, 7.14191486344257992736525376526, 7.82624745702026319644781447997, 8.255250703109015014772193500180

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