Properties

Degree 2
Conductor $ 2^{2} \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

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{20} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 20,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.470729\)
\(L(\frac12)\)  \(\approx\)  \(0.470729\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
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

−18.12718770777738096935997054363, −17.30818787808795055355503591179, −16.09956039566322976105557440854, −14.79193310953940787194244623327, −13.00236641577412261230757825497, −11.50854453196989171707939391081, −10.83473950301065015207539180976, −8.552217550781204179646223792184, −6.57891116465648258947670054106, −4.78130792717525308450176413839, 4.78130792717525308450176413839, 6.57891116465648258947670054106, 8.552217550781204179646223792184, 10.83473950301065015207539180976, 11.50854453196989171707939391081, 13.00236641577412261230757825497, 14.79193310953940787194244623327, 16.09956039566322976105557440854, 17.30818787808795055355503591179, 18.12718770777738096935997054363

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