# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.725·2-s + 19.6·3-s − 31.4·4-s − 46.7·5-s + 14.2·6-s + 49·7-s − 46.0·8-s + 143.·9-s − 33.8·10-s + 666.·11-s − 618.·12-s − 650.·13-s + 35.5·14-s − 918.·15-s + 973.·16-s + 1.18e3·17-s + 103.·18-s − 1.56e3·19-s + 1.47e3·20-s + 962.·21-s + 482.·22-s − 1.10e3·23-s − 904.·24-s − 939.·25-s − 471.·26-s − 1.96e3·27-s − 1.54e3·28-s + ⋯
 L(s)  = 1 + 0.128·2-s + 1.26·3-s − 0.983·4-s − 0.836·5-s + 0.161·6-s + 0.377·7-s − 0.254·8-s + 0.588·9-s − 0.107·10-s + 1.65·11-s − 1.23·12-s − 1.06·13-s + 0.0484·14-s − 1.05·15-s + 0.950·16-s + 0.996·17-s + 0.0754·18-s − 0.994·19-s + 0.822·20-s + 0.476·21-s + 0.212·22-s − 0.433·23-s − 0.320·24-s − 0.300·25-s − 0.136·26-s − 0.518·27-s − 0.371·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$7$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : $\chi_{7} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 7,\ (\ :5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$1.21406$$ $$L(\frac12)$$ $$\approx$$ $$1.21406$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 7$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 7$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 $$1 - 49T$$
good2 $$1 - 0.725T + 32T^{2}$$
3 $$1 - 19.6T + 243T^{2}$$
5 $$1 + 46.7T + 3.12e3T^{2}$$
11 $$1 - 666.T + 1.61e5T^{2}$$
13 $$1 + 650.T + 3.71e5T^{2}$$
17 $$1 - 1.18e3T + 1.41e6T^{2}$$
19 $$1 + 1.56e3T + 2.47e6T^{2}$$
23 $$1 + 1.10e3T + 6.43e6T^{2}$$
29 $$1 - 2.39e3T + 2.05e7T^{2}$$
31 $$1 + 2.04e3T + 2.86e7T^{2}$$
37 $$1 - 1.07e3T + 6.93e7T^{2}$$
41 $$1 - 1.09e3T + 1.15e8T^{2}$$
43 $$1 - 1.65e4T + 1.47e8T^{2}$$
47 $$1 + 8.29e3T + 2.29e8T^{2}$$
53 $$1 - 5.51e3T + 4.18e8T^{2}$$
59 $$1 + 1.42e4T + 7.14e8T^{2}$$
61 $$1 + 1.42e4T + 8.44e8T^{2}$$
67 $$1 - 1.97e4T + 1.35e9T^{2}$$
71 $$1 - 6.45e4T + 1.80e9T^{2}$$
73 $$1 - 2.85e4T + 2.07e9T^{2}$$
79 $$1 + 3.06e4T + 3.07e9T^{2}$$
83 $$1 + 675.T + 3.93e9T^{2}$$
89 $$1 - 1.25e5T + 5.58e9T^{2}$$
97 $$1 + 2.29e4T + 8.58e9T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}