Properties

 Degree 2 Conductor $2^{6} \cdot 5$ Sign $-1$ Motivic weight 5 Primitive yes Self-dual yes Analytic rank 1

Related objects

Dirichlet series

 L(s)  = 1 + 13.4·3-s + 25·5-s − 138.·7-s − 62.9·9-s + 259.·11-s − 154·13-s + 335.·15-s + 178·17-s + 965.·19-s − 1.86e3·21-s − 2.63e3·23-s + 625·25-s − 4.10e3·27-s − 4.11e3·29-s − 3.15e3·31-s + 3.48e3·33-s − 3.46e3·35-s − 7.44e3·37-s − 2.06e3·39-s + 7.27e3·41-s − 1.79e4·43-s − 1.57e3·45-s + 7.41e3·47-s + 2.41e3·49-s + 2.38e3·51-s − 3.22e4·53-s + 6.48e3·55-s + ⋯
 L(s)  = 1 + 0.860·3-s + 0.447·5-s − 1.06·7-s − 0.259·9-s + 0.646·11-s − 0.252·13-s + 0.384·15-s + 0.149·17-s + 0.613·19-s − 0.920·21-s − 1.03·23-s + 0.200·25-s − 1.08·27-s − 0.907·29-s − 0.590·31-s + 0.556·33-s − 0.478·35-s − 0.893·37-s − 0.217·39-s + 0.675·41-s − 1.47·43-s − 0.115·45-s + 0.489·47-s + 0.143·49-s + 0.128·51-s − 1.57·53-s + 0.289·55-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$320$$    =    $$2^{6} \cdot 5$$ $$\varepsilon$$ = $-1$ motivic weight = $$5$$ character : $\chi_{320} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 320,\ (\ :5/2),\ -1)$$ $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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 - 25T$$
good3 $$1 - 13.4T + 243T^{2}$$
7 $$1 + 138.T + 1.68e4T^{2}$$
11 $$1 - 259.T + 1.61e5T^{2}$$
13 $$1 + 154T + 3.71e5T^{2}$$
17 $$1 - 178T + 1.41e6T^{2}$$
19 $$1 - 965.T + 2.47e6T^{2}$$
23 $$1 + 2.63e3T + 6.43e6T^{2}$$
29 $$1 + 4.11e3T + 2.05e7T^{2}$$
31 $$1 + 3.15e3T + 2.86e7T^{2}$$
37 $$1 + 7.44e3T + 6.93e7T^{2}$$
41 $$1 - 7.27e3T + 1.15e8T^{2}$$
43 $$1 + 1.79e4T + 1.47e8T^{2}$$
47 $$1 - 7.41e3T + 2.29e8T^{2}$$
53 $$1 + 3.22e4T + 4.18e8T^{2}$$
59 $$1 - 3.40e4T + 7.14e8T^{2}$$
61 $$1 + 2.67e4T + 8.44e8T^{2}$$
67 $$1 - 4.98e4T + 1.35e9T^{2}$$
71 $$1 + 5.41e4T + 1.80e9T^{2}$$
73 $$1 + 1.85e4T + 2.07e9T^{2}$$
79 $$1 - 8.67e4T + 3.07e9T^{2}$$
83 $$1 + 7.86e4T + 3.93e9T^{2}$$
89 $$1 + 1.07e5T + 5.58e9T^{2}$$
97 $$1 + 1.08e5T + 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}