Properties

Degree 2
Conductor $ 2^{2} \cdot 5 $
Sign $1$
Motivic weight 9
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12.2·3-s − 625·5-s + 9.62e3·7-s − 1.95e4·9-s + 5.56e4·11-s + 1.69e5·13-s − 7.63e3·15-s + 2.07e5·17-s + 8.02e5·19-s + 1.17e5·21-s − 1.24e6·23-s + 3.90e5·25-s − 4.78e5·27-s − 4.28e6·29-s − 3.58e6·31-s + 6.79e5·33-s − 6.01e6·35-s − 2.89e6·37-s + 2.07e6·39-s + 2.51e7·41-s − 2.00e7·43-s + 1.22e7·45-s + 3.73e7·47-s + 5.22e7·49-s + 2.53e6·51-s − 2.55e7·53-s − 3.47e7·55-s + ⋯
L(s)  = 1  + 0.0870·3-s − 0.447·5-s + 1.51·7-s − 0.992·9-s + 1.14·11-s + 1.64·13-s − 0.0389·15-s + 0.602·17-s + 1.41·19-s + 0.131·21-s − 0.925·23-s + 0.200·25-s − 0.173·27-s − 1.12·29-s − 0.697·31-s + 0.0997·33-s − 0.677·35-s − 0.254·37-s + 0.143·39-s + 1.39·41-s − 0.893·43-s + 0.443·45-s + 1.11·47-s + 1.29·49-s + 0.0524·51-s − 0.444·53-s − 0.512·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(9\)
character  :  $\chi_{20} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 20,\ (\ :9/2),\ 1)\)
\(L(5)\)  \(\approx\)  \(1.99634\)
\(L(\frac12)\)  \(\approx\)  \(1.99634\)
\(L(\frac{11}{2})\)   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 \( 1 + 625T \)
good3 \( 1 - 12.2T + 1.96e4T^{2} \)
7 \( 1 - 9.62e3T + 4.03e7T^{2} \)
11 \( 1 - 5.56e4T + 2.35e9T^{2} \)
13 \( 1 - 1.69e5T + 1.06e10T^{2} \)
17 \( 1 - 2.07e5T + 1.18e11T^{2} \)
19 \( 1 - 8.02e5T + 3.22e11T^{2} \)
23 \( 1 + 1.24e6T + 1.80e12T^{2} \)
29 \( 1 + 4.28e6T + 1.45e13T^{2} \)
31 \( 1 + 3.58e6T + 2.64e13T^{2} \)
37 \( 1 + 2.89e6T + 1.29e14T^{2} \)
41 \( 1 - 2.51e7T + 3.27e14T^{2} \)
43 \( 1 + 2.00e7T + 5.02e14T^{2} \)
47 \( 1 - 3.73e7T + 1.11e15T^{2} \)
53 \( 1 + 2.55e7T + 3.29e15T^{2} \)
59 \( 1 + 9.96e7T + 8.66e15T^{2} \)
61 \( 1 - 2.00e8T + 1.16e16T^{2} \)
67 \( 1 + 8.09e7T + 2.72e16T^{2} \)
71 \( 1 + 4.31e7T + 4.58e16T^{2} \)
73 \( 1 + 3.40e8T + 5.88e16T^{2} \)
79 \( 1 - 2.81e8T + 1.19e17T^{2} \)
83 \( 1 + 6.01e8T + 1.86e17T^{2} \)
89 \( 1 - 5.39e8T + 3.50e17T^{2} \)
97 \( 1 - 4.23e8T + 7.60e17T^{2} \)
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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

−16.21723031146093919389283279494, −14.67269881407497906909611779116, −13.91493092406230454736814601987, −11.78884887211767794359215900147, −11.14289241511026106393967605489, −8.930418937147735166878198111736, −7.79655775891205757910738772391, −5.68085486815625618824700954297, −3.76558598593066383210248116523, −1.32686442479705478019279561910, 1.32686442479705478019279561910, 3.76558598593066383210248116523, 5.68085486815625618824700954297, 7.79655775891205757910738772391, 8.930418937147735166878198111736, 11.14289241511026106393967605489, 11.78884887211767794359215900147, 13.91493092406230454736814601987, 14.67269881407497906909611779116, 16.21723031146093919389283279494

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