# Properties

 Degree 2 Conductor $2^{2} \cdot 5$ Sign $1$ Motivic weight 11 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 542.·3-s + 3.12e3·5-s + 4.96e4·7-s + 1.16e5·9-s − 1.94e4·11-s − 7.59e5·13-s + 1.69e6·15-s + 2.35e6·17-s + 1.62e7·19-s + 2.69e7·21-s + 3.07e7·23-s + 9.76e6·25-s − 3.26e7·27-s − 1.24e8·29-s + 2.87e8·31-s − 1.05e7·33-s + 1.55e8·35-s − 6.80e8·37-s − 4.11e8·39-s − 3.20e7·41-s − 1.32e9·43-s + 3.65e8·45-s − 2.20e9·47-s + 4.88e8·49-s + 1.27e9·51-s + 4.55e9·53-s − 6.07e7·55-s + ⋯
 L(s)  = 1 + 1.28·3-s + 0.447·5-s + 1.11·7-s + 0.660·9-s − 0.0364·11-s − 0.567·13-s + 0.576·15-s + 0.402·17-s + 1.50·19-s + 1.43·21-s + 0.997·23-s + 0.199·25-s − 0.437·27-s − 1.12·29-s + 1.80·31-s − 0.0469·33-s + 0.499·35-s − 1.61·37-s − 0.731·39-s − 0.0432·41-s − 1.37·43-s + 0.295·45-s − 1.40·47-s + 0.247·49-s + 0.518·51-s + 1.49·53-s − 0.0162·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$20$$    =    $$2^{2} \cdot 5$$ $$\varepsilon$$ = $1$ motivic weight = $$11$$ character : $\chi_{20} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 20,\ (\ :11/2),\ 1)$ $L(6)$ $\approx$ $3.29744$ $L(\frac12)$ $\approx$ $3.29744$ $L(\frac{13}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
5 $$1 - 3.12e3T$$
good3 $$1 - 542.T + 1.77e5T^{2}$$
7 $$1 - 4.96e4T + 1.97e9T^{2}$$
11 $$1 + 1.94e4T + 2.85e11T^{2}$$
13 $$1 + 7.59e5T + 1.79e12T^{2}$$
17 $$1 - 2.35e6T + 3.42e13T^{2}$$
19 $$1 - 1.62e7T + 1.16e14T^{2}$$
23 $$1 - 3.07e7T + 9.52e14T^{2}$$
29 $$1 + 1.24e8T + 1.22e16T^{2}$$
31 $$1 - 2.87e8T + 2.54e16T^{2}$$
37 $$1 + 6.80e8T + 1.77e17T^{2}$$
41 $$1 + 3.20e7T + 5.50e17T^{2}$$
43 $$1 + 1.32e9T + 9.29e17T^{2}$$
47 $$1 + 2.20e9T + 2.47e18T^{2}$$
53 $$1 - 4.55e9T + 9.26e18T^{2}$$
59 $$1 + 1.70e9T + 3.01e19T^{2}$$
61 $$1 + 6.50e9T + 4.35e19T^{2}$$
67 $$1 - 6.87e7T + 1.22e20T^{2}$$
71 $$1 + 2.55e10T + 2.31e20T^{2}$$
73 $$1 - 1.19e10T + 3.13e20T^{2}$$
79 $$1 + 9.68e9T + 7.47e20T^{2}$$
83 $$1 + 1.07e10T + 1.28e21T^{2}$$
89 $$1 + 7.08e10T + 2.77e21T^{2}$$
97 $$1 + 1.53e10T + 7.15e21T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}