Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 272.·3-s − 625·5-s − 1.00e4·7-s + 5.44e4·9-s + 4.70e4·11-s + 9.36e3·13-s + 1.70e5·15-s + 1.08e5·17-s − 6.65e5·19-s + 2.72e6·21-s + 5.76e5·23-s + 3.90e5·25-s − 9.45e6·27-s − 2.61e6·29-s + 3.87e6·31-s − 1.28e7·33-s + 6.25e6·35-s + 1.41e7·37-s − 2.54e6·39-s + 4.62e6·41-s + 8.31e6·43-s − 3.40e7·45-s + 2.51e7·47-s + 5.96e7·49-s − 2.95e7·51-s + 3.49e7·53-s − 2.94e7·55-s + ⋯
L(s)  = 1  − 1.94·3-s − 0.447·5-s − 1.57·7-s + 2.76·9-s + 0.969·11-s + 0.0909·13-s + 0.867·15-s + 0.315·17-s − 1.17·19-s + 3.05·21-s + 0.429·23-s + 0.200·25-s − 3.42·27-s − 0.685·29-s + 0.754·31-s − 1.88·33-s + 0.704·35-s + 1.24·37-s − 0.176·39-s + 0.255·41-s + 0.370·43-s − 1.23·45-s + 0.753·47-s + 1.47·49-s − 0.611·51-s + 0.608·53-s − 0.433·55-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  =  \((2,\ 20,\ (\ :9/2),\ 1)\)
\(L(5)\)  \(\approx\)  \(0.558347\)
\(L(\frac12)\)  \(\approx\)  \(0.558347\)
\(L(\frac{11}{2})\)   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 \( 1 + 272.T + 1.96e4T^{2} \)
7 \( 1 + 1.00e4T + 4.03e7T^{2} \)
11 \( 1 - 4.70e4T + 2.35e9T^{2} \)
13 \( 1 - 9.36e3T + 1.06e10T^{2} \)
17 \( 1 - 1.08e5T + 1.18e11T^{2} \)
19 \( 1 + 6.65e5T + 3.22e11T^{2} \)
23 \( 1 - 5.76e5T + 1.80e12T^{2} \)
29 \( 1 + 2.61e6T + 1.45e13T^{2} \)
31 \( 1 - 3.87e6T + 2.64e13T^{2} \)
37 \( 1 - 1.41e7T + 1.29e14T^{2} \)
41 \( 1 - 4.62e6T + 3.27e14T^{2} \)
43 \( 1 - 8.31e6T + 5.02e14T^{2} \)
47 \( 1 - 2.51e7T + 1.11e15T^{2} \)
53 \( 1 - 3.49e7T + 3.29e15T^{2} \)
59 \( 1 - 6.71e6T + 8.66e15T^{2} \)
61 \( 1 + 4.75e6T + 1.16e16T^{2} \)
67 \( 1 + 1.38e8T + 2.72e16T^{2} \)
71 \( 1 - 3.54e8T + 4.58e16T^{2} \)
73 \( 1 - 2.41e8T + 5.88e16T^{2} \)
79 \( 1 - 2.61e8T + 1.19e17T^{2} \)
83 \( 1 + 6.55e8T + 1.86e17T^{2} \)
89 \( 1 + 1.00e9T + 3.50e17T^{2} \)
97 \( 1 - 1.24e9T + 7.60e17T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.51122131683281429736759222612, −15.40185298097005058775829723939, −12.97609545859416210930129435340, −12.13656968920087926388401352623, −10.92628573734094939785313602525, −9.639685230494375499020864739510, −6.89071308966658125369361898257, −5.99825631602642481469719988168, −4.10967824369615132705621681419, −0.65104259923725667941952006804, 0.65104259923725667941952006804, 4.10967824369615132705621681419, 5.99825631602642481469719988168, 6.89071308966658125369361898257, 9.639685230494375499020864739510, 10.92628573734094939785313602525, 12.13656968920087926388401352623, 12.97609545859416210930129435340, 15.40185298097005058775829723939, 16.51122131683281429736759222612

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