Properties

Degree 2
Conductor $ 2^{4} \cdot 5 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s − 2·7-s + 9-s + 2·13-s − 2·15-s − 6·17-s + 4·19-s − 4·21-s − 6·23-s + 25-s − 4·27-s + 6·29-s + 4·31-s + 2·35-s + 2·37-s + 4·39-s + 6·41-s + 10·43-s − 45-s + 6·47-s − 3·49-s − 12·51-s − 6·53-s + 8·57-s − 12·59-s + 2·61-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.917·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.338·35-s + 0.328·37-s + 0.640·39-s + 0.937·41-s + 1.52·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 1.68·51-s − 0.824·53-s + 1.05·57-s − 1.56·59-s + 0.256·61-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(80\)    =    \(2^{4} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{80} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 80,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.137082599$
$L(\frac12)$  $\approx$  $1.137082599$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.96845653879085, −19.68269158036047, −18.59554052210670, −17.52000693171683, −15.92795853427658, −15.61700859461542, −14.20933451643818, −13.57191933746624, −12.41085901163163, −11.10973086408237, −9.683565298540966, −8.740293324354028, −7.712731108234995, −6.270876241928756, −4.122504333686863, −2.769298906172612, 2.769298906172612, 4.122504333686863, 6.270876241928756, 7.712731108234995, 8.740293324354028, 9.683565298540966, 11.10973086408237, 12.41085901163163, 13.57191933746624, 14.20933451643818, 15.61700859461542, 15.92795853427658, 17.52000693171683, 18.59554052210670, 19.68269158036047, 19.96845653879085

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