Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.995 - 0.0941i$
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 + (0.142 − 0.278i)13-s + (−0.278 + 1.76i)17-s + (1.11 − 1.53i)29-s + (0.809 + 0.412i)37-s + (0.363 + 1.11i)41-s + i·49-s + (−0.309 − 1.95i)53-s + (−0.363 + 1.11i)61-s + (1.76 − 0.896i)73-s + (0.809 − 0.587i)81-s + (−1.80 − 0.587i)89-s + (−0.896 + 0.142i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)9-s + (0.142 − 0.278i)13-s + (−0.278 + 1.76i)17-s + (1.11 − 1.53i)29-s + (0.809 + 0.412i)37-s + (0.363 + 1.11i)41-s + i·49-s + (−0.309 − 1.95i)53-s + (−0.363 + 1.11i)61-s + (1.76 − 0.896i)73-s + (0.809 − 0.587i)81-s + (−1.80 − 0.587i)89-s + (−0.896 + 0.142i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.995 - 0.0941i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (993, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ 0.995 - 0.0941i)$
$L(\frac{1}{2})$  $\approx$  $1.433158447$
$L(\frac12)$  $\approx$  $1.433158447$
$L(1)$   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 \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \)
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 \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \)
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 \( 1 + (0.309 + 1.95i)T + (-0.951 + 0.309i)T^{2} \)
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 \( 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2} \)
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 \( 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2} \)
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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

−8.356189308896358762033099074657, −8.124848606683721861021854230033, −7.14471488424670489859552369971, −6.32329636484908448427440554272, −5.92860275214370784695192142568, −4.65057092061303244437634668960, −4.17739233530411504358814569038, −3.26671263641733394939534914498, −2.14314490662034437229156457093, −1.12404826424890290748201183915, 1.03929345663696397723061347531, 2.21338227684567102175597901696, 3.10644174595191372097002541949, 4.16855084994953240017632508608, 4.82555032191428204069120998304, 5.51865993930006968129874883538, 6.67102674378851092570350481336, 7.07696883134046784489884065113, 7.77813304337896889801104315465, 8.684177896088340806778709659554

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