# Properties

 Degree 2 Conductor $2^{6} \cdot 5$ Sign $-1$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 7.81i·3-s − 55.9·5-s + 171. i·7-s + 181.·9-s + 437. i·15-s + 1.34e3·21-s − 230. i·23-s + 3.12e3·25-s − 3.32e3i·27-s − 1.68e3·29-s − 9.60e3i·35-s − 2.10e4·41-s − 2.20e4i·43-s − 1.01e4·45-s + 3.01e4i·47-s + ⋯
 L(s)  = 1 − 0.501i·3-s − 0.999·5-s + 1.32i·7-s + 0.748·9-s + 0.501i·15-s + 0.664·21-s − 0.0909i·23-s + 25-s − 0.877i·27-s − 0.372·29-s − 1.32i·35-s − 1.95·41-s − 1.81i·43-s − 0.748·45-s + 1.98i·47-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$320$$    =    $$2^{6} \cdot 5$$ $$\varepsilon$$ = $-1$ motivic weight = $$5$$ character : $\chi_{320} (129, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 320,\ (\ :5/2),\ -1)$$ $$L(3)$$ $$\approx$$ $$0.05341236993$$ $$L(\frac12)$$ $$\approx$$ $$0.05341236993$$ $$L(\frac{7}{2})$$ 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 + 55.9T$$
good3 $$1 + 7.81iT - 243T^{2}$$
7 $$1 - 171. iT - 1.68e4T^{2}$$
11 $$1 + 1.61e5T^{2}$$
13 $$1 - 3.71e5T^{2}$$
17 $$1 - 1.41e6T^{2}$$
19 $$1 + 2.47e6T^{2}$$
23 $$1 + 230. iT - 6.43e6T^{2}$$
29 $$1 + 1.68e3T + 2.05e7T^{2}$$
31 $$1 + 2.86e7T^{2}$$
37 $$1 - 6.93e7T^{2}$$
41 $$1 + 2.10e4T + 1.15e8T^{2}$$
43 $$1 + 2.20e4iT - 1.47e8T^{2}$$
47 $$1 - 3.01e4iT - 2.29e8T^{2}$$
53 $$1 - 4.18e8T^{2}$$
59 $$1 + 7.14e8T^{2}$$
61 $$1 + 5.22e4T + 8.44e8T^{2}$$
67 $$1 - 3.68e4iT - 1.35e9T^{2}$$
71 $$1 + 1.80e9T^{2}$$
73 $$1 - 2.07e9T^{2}$$
79 $$1 + 3.07e9T^{2}$$
83 $$1 + 4.67e4iT - 3.93e9T^{2}$$
89 $$1 + 1.49e5T + 5.58e9T^{2}$$
97 $$1 - 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}