Properties

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

Origins

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Normalization:  

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

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