Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.443 - 0.896i$
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 + (−0.896 + 1.76i)13-s + (−0.142 + 0.896i)17-s + (−1.11 + 1.53i)29-s + (1.58 + 0.809i)37-s + (−0.363 − 1.11i)41-s + i·49-s + (0.0489 + 0.309i)53-s + (0.363 − 1.11i)61-s + (−0.278 + 0.142i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (−1.76 + 0.278i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)9-s + (−0.896 + 1.76i)13-s + (−0.142 + 0.896i)17-s + (−1.11 + 1.53i)29-s + (1.58 + 0.809i)37-s + (−0.363 − 1.11i)41-s + i·49-s + (0.0489 + 0.309i)53-s + (0.363 − 1.11i)61-s + (−0.278 + 0.142i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (−1.76 + 0.278i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.443 - 0.896i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (993, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ 0.443 - 0.896i)$
$L(\frac{1}{2})$  $\approx$  $1.215523878$
$L(\frac12)$  $\approx$  $1.215523878$
$L(1)$   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 \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \)
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 \( 1 + (-1.58 - 0.809i)T + (0.587 + 0.809i)T^{2} \)
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 \( 1 + (-0.0489 - 0.309i)T + (-0.951 + 0.309i)T^{2} \)
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 \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \)
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 \( 1 + (1.76 - 0.278i)T + (0.951 - 0.309i)T^{2} \)
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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

−8.936109210704332995079557041838, −7.946992168947349799146559467082, −7.13967258695315259701561422407, −6.74219191672740514947373011864, −5.88875441566962487467539298371, −4.81898593312863211863415280434, −4.26487735037245109466832037990, −3.48444567854195544248087335568, −2.19462611321061363461237086159, −1.45272634367953630068183288294, 0.69952285132200704004475906738, 2.14485206147288123826099364059, 2.90822105014798316506708198786, 3.96557487189154611714231957546, 4.78257352487981085102458327090, 5.45352889028112300227393529523, 6.24415856981410485079688219314, 7.34582093932608177743243266760, 7.59578698862418475996734537593, 8.325951245511317579203344336193

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