# Properties

 Degree 2 Conductor $2^{4} \cdot 5$ Sign $-1$ Motivic weight 5 Primitive yes Self-dual yes Analytic rank 1

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## Dirichlet series

 L(s)  = 1 − 24·3-s + 25·5-s + 172·7-s + 333·9-s − 132·11-s − 946·13-s − 600·15-s − 222·17-s − 500·19-s − 4.12e3·21-s − 3.56e3·23-s + 625·25-s − 2.16e3·27-s + 2.19e3·29-s − 2.31e3·31-s + 3.16e3·33-s + 4.30e3·35-s − 1.12e4·37-s + 2.27e4·39-s + 1.24e3·41-s − 2.06e4·43-s + 8.32e3·45-s − 6.58e3·47-s + 1.27e4·49-s + 5.32e3·51-s − 2.10e4·53-s − 3.30e3·55-s + ⋯
 L(s)  = 1 − 1.53·3-s + 0.447·5-s + 1.32·7-s + 1.37·9-s − 0.328·11-s − 1.55·13-s − 0.688·15-s − 0.186·17-s − 0.317·19-s − 2.04·21-s − 1.40·23-s + 1/5·25-s − 0.570·27-s + 0.483·29-s − 0.432·31-s + 0.506·33-s + 0.593·35-s − 1.35·37-s + 2.39·39-s + 0.115·41-s − 1.70·43-s + 0.612·45-s − 0.435·47-s + 0.760·49-s + 0.286·51-s − 1.03·53-s − 0.147·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$80$$    =    $$2^{4} \cdot 5$$ $$\varepsilon$$ = $-1$ motivic weight = $$5$$ character : $\chi_{80} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 80,\ (\ :5/2),\ -1)$$ $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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 - p^{2} T$$
good3 $$1 + 8 p T + p^{5} T^{2}$$
7 $$1 - 172 T + p^{5} T^{2}$$
11 $$1 + 12 p T + p^{5} T^{2}$$
13 $$1 + 946 T + p^{5} T^{2}$$
17 $$1 + 222 T + p^{5} T^{2}$$
19 $$1 + 500 T + p^{5} T^{2}$$
23 $$1 + 3564 T + p^{5} T^{2}$$
29 $$1 - 2190 T + p^{5} T^{2}$$
31 $$1 + 2312 T + p^{5} T^{2}$$
37 $$1 + 11242 T + p^{5} T^{2}$$
41 $$1 - 1242 T + p^{5} T^{2}$$
43 $$1 + 20624 T + p^{5} T^{2}$$
47 $$1 + 6588 T + p^{5} T^{2}$$
53 $$1 + 21066 T + p^{5} T^{2}$$
59 $$1 + 7980 T + p^{5} T^{2}$$
61 $$1 - 16622 T + p^{5} T^{2}$$
67 $$1 + 1808 T + p^{5} T^{2}$$
71 $$1 - 24528 T + p^{5} T^{2}$$
73 $$1 - 20474 T + p^{5} T^{2}$$
79 $$1 - 46240 T + p^{5} T^{2}$$
83 $$1 - 51576 T + p^{5} T^{2}$$
89 $$1 + 110310 T + p^{5} T^{2}$$
97 $$1 + 78382 T + p^{5} 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

−12.47995470179743287008309492312, −11.77288162308099442065404139951, −10.78288945001328173769090173844, −9.891011945437266654294936196237, −8.086835065811471648783639670531, −6.75833492559590758017683165002, −5.39977897845645117379406857193, −4.70261232999786746412203567994, −1.86359351139635476473656660725, 0, 1.86359351139635476473656660725, 4.70261232999786746412203567994, 5.39977897845645117379406857193, 6.75833492559590758017683165002, 8.086835065811471648783639670531, 9.891011945437266654294936196237, 10.78288945001328173769090173844, 11.77288162308099442065404139951, 12.47995470179743287008309492312