# Properties

 Label 2-7e2-49.8-c3-0-11 Degree $2$ Conductor $49$ Sign $0.833 + 0.552i$ Analytic cond. $2.89109$ Root an. cond. $1.70032$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.521 + 0.654i)2-s + (5.78 − 2.78i)3-s + (1.62 − 7.11i)4-s + (−1.28 + 0.620i)5-s + (4.83 + 2.32i)6-s + (−17.3 − 6.46i)7-s + (11.5 − 5.55i)8-s + (8.82 − 11.0i)9-s + (−1.07 − 0.519i)10-s + (33.8 + 42.5i)11-s + (−10.4 − 45.6i)12-s + (41.8 + 52.5i)13-s + (−4.82 − 14.7i)14-s + (−5.72 + 7.17i)15-s + (−42.9 − 20.6i)16-s + (−4.21 − 18.4i)17-s + ⋯
 L(s)  = 1 + (0.184 + 0.231i)2-s + (1.11 − 0.535i)3-s + (0.203 − 0.889i)4-s + (−0.115 + 0.0555i)5-s + (0.329 + 0.158i)6-s + (−0.937 − 0.349i)7-s + (0.509 − 0.245i)8-s + (0.326 − 0.410i)9-s + (−0.0341 − 0.0164i)10-s + (0.929 + 1.16i)11-s + (−0.250 − 1.09i)12-s + (0.893 + 1.12i)13-s + (−0.0921 − 0.281i)14-s + (−0.0985 + 0.123i)15-s + (−0.671 − 0.323i)16-s + (−0.0600 − 0.263i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$49$$    =    $$7^{2}$$ Sign: $0.833 + 0.552i$ Analytic conductor: $$2.89109$$ Root analytic conductor: $$1.70032$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{49} (8, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 49,\ (\ :3/2),\ 0.833 + 0.552i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.87407 - 0.564156i$$ $$L(\frac12)$$ $$\approx$$ $$1.87407 - 0.564156i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (17.3 + 6.46i)T$$
good2 $$1 + (-0.521 - 0.654i)T + (-1.78 + 7.79i)T^{2}$$
3 $$1 + (-5.78 + 2.78i)T + (16.8 - 21.1i)T^{2}$$
5 $$1 + (1.28 - 0.620i)T + (77.9 - 97.7i)T^{2}$$
11 $$1 + (-33.8 - 42.5i)T + (-296. + 1.29e3i)T^{2}$$
13 $$1 + (-41.8 - 52.5i)T + (-488. + 2.14e3i)T^{2}$$
17 $$1 + (4.21 + 18.4i)T + (-4.42e3 + 2.13e3i)T^{2}$$
19 $$1 + 58.6T + 6.85e3T^{2}$$
23 $$1 + (4.27 - 18.7i)T + (-1.09e4 - 5.27e3i)T^{2}$$
29 $$1 + (50.8 + 222. i)T + (-2.19e4 + 1.05e4i)T^{2}$$
31 $$1 - 165.T + 2.97e4T^{2}$$
37 $$1 + (53.1 + 232. i)T + (-4.56e4 + 2.19e4i)T^{2}$$
41 $$1 + (174. - 84.1i)T + (4.29e4 - 5.38e4i)T^{2}$$
43 $$1 + (206. + 99.2i)T + (4.95e4 + 6.21e4i)T^{2}$$
47 $$1 + (-153. - 192. i)T + (-2.31e4 + 1.01e5i)T^{2}$$
53 $$1 + (-118. + 518. i)T + (-1.34e5 - 6.45e4i)T^{2}$$
59 $$1 + (150. + 72.6i)T + (1.28e5 + 1.60e5i)T^{2}$$
61 $$1 + (-100. - 441. i)T + (-2.04e5 + 9.84e4i)T^{2}$$
67 $$1 - 153.T + 3.00e5T^{2}$$
71 $$1 + (-137. + 603. i)T + (-3.22e5 - 1.55e5i)T^{2}$$
73 $$1 + (247. - 310. i)T + (-8.65e4 - 3.79e5i)T^{2}$$
79 $$1 + 981.T + 4.93e5T^{2}$$
83 $$1 + (258. - 323. i)T + (-1.27e5 - 5.57e5i)T^{2}$$
89 $$1 + (-833. + 1.04e3i)T + (-1.56e5 - 6.87e5i)T^{2}$$
97 $$1 - 1.06e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$