# 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

# Related objects

## 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