# Properties

 Degree 2 Conductor 5 Sign $1$ Motivic weight 7 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.28·2-s + 79.7·3-s − 126.·4-s − 125·5-s + 102.·6-s − 538.·7-s − 326.·8-s + 4.17e3·9-s − 160.·10-s − 1.21e3·11-s − 1.00e4·12-s + 7.07e3·13-s − 690.·14-s − 9.96e3·15-s + 1.57e4·16-s − 3.34e3·17-s + 5.34e3·18-s + 2.21e4·19-s + 1.57e4·20-s − 4.29e4·21-s − 1.55e3·22-s − 5.85e4·23-s − 2.60e4·24-s + 1.56e4·25-s + 9.06e3·26-s + 1.58e5·27-s + 6.80e4·28-s + ⋯
 L(s)  = 1 + 0.113·2-s + 1.70·3-s − 0.987·4-s − 0.447·5-s + 0.193·6-s − 0.593·7-s − 0.225·8-s + 1.90·9-s − 0.0506·10-s − 0.275·11-s − 1.68·12-s + 0.892·13-s − 0.0672·14-s − 0.762·15-s + 0.961·16-s − 0.165·17-s + 0.216·18-s + 0.741·19-s + 0.441·20-s − 1.01·21-s − 0.0311·22-s − 1.00·23-s − 0.384·24-s + 0.199·25-s + 0.101·26-s + 1.54·27-s + 0.585·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$5$$ $$\varepsilon$$ = $1$ motivic weight = $$7$$ character : $\chi_{5} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 5,\ (\ :7/2),\ 1)$ $L(4)$ $\approx$ $1.46766$ $L(\frac12)$ $\approx$ $1.46766$ $L(\frac{9}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $$F_p$$ is a polynomial of degree 2. If $p = 5$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 $$1 + 125T$$
good2 $$1 - 1.28T + 128T^{2}$$
3 $$1 - 79.7T + 2.18e3T^{2}$$
7 $$1 + 538.T + 8.23e5T^{2}$$
11 $$1 + 1.21e3T + 1.94e7T^{2}$$
13 $$1 - 7.07e3T + 6.27e7T^{2}$$
17 $$1 + 3.34e3T + 4.10e8T^{2}$$
19 $$1 - 2.21e4T + 8.93e8T^{2}$$
23 $$1 + 5.85e4T + 3.40e9T^{2}$$
29 $$1 + 2.06e5T + 1.72e10T^{2}$$
31 $$1 - 1.77e5T + 2.75e10T^{2}$$
37 $$1 + 2.84e5T + 9.49e10T^{2}$$
41 $$1 - 6.27e5T + 1.94e11T^{2}$$
43 $$1 + 1.64e5T + 2.71e11T^{2}$$
47 $$1 + 4.49e5T + 5.06e11T^{2}$$
53 $$1 + 7.30e5T + 1.17e12T^{2}$$
59 $$1 - 1.42e6T + 2.48e12T^{2}$$
61 $$1 + 2.66e5T + 3.14e12T^{2}$$
67 $$1 - 2.95e6T + 6.06e12T^{2}$$
71 $$1 - 9.21e5T + 9.09e12T^{2}$$
73 $$1 - 4.25e6T + 1.10e13T^{2}$$
79 $$1 - 6.28e6T + 1.92e13T^{2}$$
83 $$1 + 9.17e6T + 2.71e13T^{2}$$
89 $$1 - 2.42e5T + 4.42e13T^{2}$$
97 $$1 + 2.59e6T + 8.07e13T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}