# Properties

 Label 2-7e2-49.25-c5-0-4 Degree $2$ Conductor $49$ Sign $0.862 - 0.505i$ Analytic cond. $7.85880$ Root an. cond. $2.80335$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.66 + 0.822i)2-s + (−6.33 − 16.1i)3-s + (−20.0 + 13.6i)4-s + (−58.2 − 8.78i)5-s + (30.1 + 37.8i)6-s + (54.7 + 117. i)7-s + (97.7 − 122. i)8-s + (−42.5 + 39.4i)9-s + (162. − 24.5i)10-s + (260. + 242. i)11-s + (347. + 236. i)12-s + (210. − 922. i)13-s + (−242. − 268. i)14-s + (227. + 997. i)15-s + (123. − 313. i)16-s + (−98.0 + 1.30e3i)17-s + ⋯
 L(s)  = 1 + (−0.471 + 0.145i)2-s + (−0.406 − 1.03i)3-s + (−0.625 + 0.426i)4-s + (−1.04 − 0.157i)5-s + (0.342 + 0.429i)6-s + (0.422 + 0.906i)7-s + (0.540 − 0.677i)8-s + (−0.175 + 0.162i)9-s + (0.514 − 0.0775i)10-s + (0.650 + 0.603i)11-s + (0.695 + 0.474i)12-s + (0.345 − 1.51i)13-s + (−0.330 − 0.365i)14-s + (0.261 + 1.14i)15-s + (0.120 − 0.306i)16-s + (−0.0822 + 1.09i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$49$$    =    $$7^{2}$$ Sign: $0.862 - 0.505i$ Analytic conductor: $$7.85880$$ Root analytic conductor: $$2.80335$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{49} (25, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 49,\ (\ :5/2),\ 0.862 - 0.505i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.726961 + 0.197337i$$ $$L(\frac12)$$ $$\approx$$ $$0.726961 + 0.197337i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (-54.7 - 117. i)T$$
good2 $$1 + (2.66 - 0.822i)T + (26.4 - 18.0i)T^{2}$$
3 $$1 + (6.33 + 16.1i)T + (-178. + 165. i)T^{2}$$
5 $$1 + (58.2 + 8.78i)T + (2.98e3 + 921. i)T^{2}$$
11 $$1 + (-260. - 242. i)T + (1.20e4 + 1.60e5i)T^{2}$$
13 $$1 + (-210. + 922. i)T + (-3.34e5 - 1.61e5i)T^{2}$$
17 $$1 + (98.0 - 1.30e3i)T + (-1.40e6 - 2.11e5i)T^{2}$$
19 $$1 + (-1.28e3 - 2.22e3i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (-250. - 3.33e3i)T + (-6.36e6 + 9.59e5i)T^{2}$$
29 $$1 + (1.35e3 - 652. i)T + (1.27e7 - 1.60e7i)T^{2}$$
31 $$1 + (-4.22e3 + 7.31e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (-6.62e3 - 4.51e3i)T + (2.53e7 + 6.45e7i)T^{2}$$
41 $$1 + (-5.96e3 + 7.48e3i)T + (-2.57e7 - 1.12e8i)T^{2}$$
43 $$1 + (-1.21e4 - 1.51e4i)T + (-3.27e7 + 1.43e8i)T^{2}$$
47 $$1 + (8.53e3 - 2.63e3i)T + (1.89e8 - 1.29e8i)T^{2}$$
53 $$1 + (8.33e3 - 5.67e3i)T + (1.52e8 - 3.89e8i)T^{2}$$
59 $$1 + (-3.38e4 + 5.09e3i)T + (6.83e8 - 2.10e8i)T^{2}$$
61 $$1 + (-3.12e4 - 2.12e4i)T + (3.08e8 + 7.86e8i)T^{2}$$
67 $$1 + (-5.99e3 + 1.03e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + (2.54e4 + 1.22e4i)T + (1.12e9 + 1.41e9i)T^{2}$$
73 $$1 + (-1.60e4 - 4.94e3i)T + (1.71e9 + 1.16e9i)T^{2}$$
79 $$1 + (1.05e3 + 1.83e3i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + (-527. - 2.31e3i)T + (-3.54e9 + 1.70e9i)T^{2}$$
89 $$1 + (2.46e4 - 2.28e4i)T + (4.17e8 - 5.56e9i)T^{2}$$
97 $$1 + 5.46e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$