# Properties

 Label 2-7e2-49.2-c5-0-2 Degree $2$ Conductor $49$ Sign $0.750 - 0.660i$ 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 + (−10.0 − 3.09i)2-s + (9.56 − 24.3i)3-s + (64.7 + 44.1i)4-s + (−25.7 + 3.88i)5-s + (−171. + 214. i)6-s + (24.4 + 127. i)7-s + (−303. − 381. i)8-s + (−323. − 300. i)9-s + (270. + 40.8i)10-s + (−479. + 444. i)11-s + (1.69e3 − 1.15e3i)12-s + (195. + 855. i)13-s + (148. − 1.35e3i)14-s + (−151. + 665. i)15-s + (954. + 2.43e3i)16-s + (59.6 + 795. i)17-s + ⋯
 L(s)  = 1 + (−1.77 − 0.547i)2-s + (0.613 − 1.56i)3-s + (2.02 + 1.37i)4-s + (−0.461 + 0.0695i)5-s + (−1.94 + 2.43i)6-s + (0.188 + 0.981i)7-s + (−1.67 − 2.10i)8-s + (−1.33 − 1.23i)9-s + (0.856 + 0.129i)10-s + (−1.19 + 1.10i)11-s + (3.39 − 2.31i)12-s + (0.320 + 1.40i)13-s + (0.202 − 1.84i)14-s + (−0.174 + 0.763i)15-s + (0.932 + 2.37i)16-s + (0.0500 + 0.667i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$49$$    =    $$7^{2}$$ Sign: $0.750 - 0.660i$ Analytic conductor: $$7.85880$$ Root analytic conductor: $$2.80335$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{49} (2, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 49,\ (\ :5/2),\ 0.750 - 0.660i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.375858 + 0.141907i$$ $$L(\frac12)$$ $$\approx$$ $$0.375858 + 0.141907i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (-24.4 - 127. i)T$$
good2 $$1 + (10.0 + 3.09i)T + (26.4 + 18.0i)T^{2}$$
3 $$1 + (-9.56 + 24.3i)T + (-178. - 165. i)T^{2}$$
5 $$1 + (25.7 - 3.88i)T + (2.98e3 - 921. i)T^{2}$$
11 $$1 + (479. - 444. i)T + (1.20e4 - 1.60e5i)T^{2}$$
13 $$1 + (-195. - 855. i)T + (-3.34e5 + 1.61e5i)T^{2}$$
17 $$1 + (-59.6 - 795. i)T + (-1.40e6 + 2.11e5i)T^{2}$$
19 $$1 + (-702. + 1.21e3i)T + (-1.23e6 - 2.14e6i)T^{2}$$
23 $$1 + (-68.6 + 916. i)T + (-6.36e6 - 9.59e5i)T^{2}$$
29 $$1 + (675. + 325. i)T + (1.27e7 + 1.60e7i)T^{2}$$
31 $$1 + (-18.1 - 31.3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + (-132. + 90.4i)T + (2.53e7 - 6.45e7i)T^{2}$$
41 $$1 + (-5.24e3 - 6.57e3i)T + (-2.57e7 + 1.12e8i)T^{2}$$
43 $$1 + (6.76e3 - 8.48e3i)T + (-3.27e7 - 1.43e8i)T^{2}$$
47 $$1 + (1.24e4 + 3.83e3i)T + (1.89e8 + 1.29e8i)T^{2}$$
53 $$1 + (1.22e4 + 8.37e3i)T + (1.52e8 + 3.89e8i)T^{2}$$
59 $$1 + (-2.58e4 - 3.89e3i)T + (6.83e8 + 2.10e8i)T^{2}$$
61 $$1 + (2.29e4 - 1.56e4i)T + (3.08e8 - 7.86e8i)T^{2}$$
67 $$1 + (-3.00e4 - 5.20e4i)T + (-6.75e8 + 1.16e9i)T^{2}$$
71 $$1 + (1.70e4 - 8.21e3i)T + (1.12e9 - 1.41e9i)T^{2}$$
73 $$1 + (2.02e3 - 624. i)T + (1.71e9 - 1.16e9i)T^{2}$$
79 $$1 + (-3.91e4 + 6.77e4i)T + (-1.53e9 - 2.66e9i)T^{2}$$
83 $$1 + (1.72e4 - 7.55e4i)T + (-3.54e9 - 1.70e9i)T^{2}$$
89 $$1 + (-3.37e4 - 3.13e4i)T + (4.17e8 + 5.56e9i)T^{2}$$
97 $$1 + 9.73e4T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$