# Properties

 Degree 2 Conductor 7 Sign $0.120 + 0.992i$ Motivic weight 5 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−3.54 − 6.13i)2-s + (5.04 − 8.73i)3-s + (−9.08 + 15.7i)4-s + (39.9 + 69.1i)5-s − 71.4·6-s + (43.1 − 122. i)7-s − 97.9·8-s + (70.6 + 122. i)9-s + (282. − 489. i)10-s + (−175. + 304. i)11-s + (91.5 + 158. i)12-s − 291.·13-s + (−902. + 168. i)14-s + 804.·15-s + (637. + 1.10e3i)16-s + (185. − 320. i)17-s + ⋯
 L(s)  = 1 + (−0.626 − 1.08i)2-s + (0.323 − 0.560i)3-s + (−0.283 + 0.491i)4-s + (0.713 + 1.23i)5-s − 0.809·6-s + (0.332 − 0.942i)7-s − 0.541·8-s + (0.290 + 0.503i)9-s + (0.893 − 1.54i)10-s + (−0.438 + 0.759i)11-s + (0.183 + 0.317i)12-s − 0.478·13-s + (−1.23 + 0.229i)14-s + 0.923·15-s + (0.622 + 1.07i)16-s + (0.155 − 0.268i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$7$$ $$\varepsilon$$ = $0.120 + 0.992i$ motivic weight = $$5$$ character : $\chi_{7} (4, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 7,\ (\ :5/2),\ 0.120 + 0.992i)$ $L(3)$ $\approx$ $0.697334 - 0.617778i$ $L(\frac12)$ $\approx$ $0.697334 - 0.617778i$ $L(\frac{7}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 7$, $$F_p$$ is a polynomial of degree 2. If $p = 7$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 $$1 + (-43.1 + 122. i)T$$
good2 $$1 + (3.54 + 6.13i)T + (-16 + 27.7i)T^{2}$$
3 $$1 + (-5.04 + 8.73i)T + (-121.5 - 210. i)T^{2}$$
5 $$1 + (-39.9 - 69.1i)T + (-1.56e3 + 2.70e3i)T^{2}$$
11 $$1 + (175. - 304. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + 291.T + 3.71e5T^{2}$$
17 $$1 + (-185. + 320. i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (752. + 1.30e3i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (-212. - 368. i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + 7.78e3T + 2.05e7T^{2}$$
31 $$1 + (-1.28e3 + 2.23e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (369. + 640. i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 - 7.02e3T + 1.15e8T^{2}$$
43 $$1 - 1.83e3T + 1.47e8T^{2}$$
47 $$1 + (-766. - 1.32e3i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (-4.76e3 + 8.25e3i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (-1.48e4 + 2.56e4i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (-2.32e4 - 4.02e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (1.33e4 - 2.31e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 1.43e4T + 1.80e9T^{2}$$
73 $$1 + (-3.50e4 + 6.07e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (-1.35e4 - 2.34e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + 7.97e4T + 3.93e9T^{2}$$
89 $$1 + (2.17e4 + 3.77e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 - 1.03e5T + 8.58e9T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}