# Properties

 Label 2-7e2-49.15-c3-0-7 Degree $2$ Conductor $49$ Sign $0.932 + 0.360i$ 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 + (0.0613 − 0.268i)2-s + (2.25 − 2.82i)3-s + (7.13 + 3.43i)4-s + (−2.36 + 2.97i)5-s + (−0.620 − 0.778i)6-s + (9.47 − 15.9i)7-s + (2.73 − 3.43i)8-s + (3.10 + 13.6i)9-s + (0.653 + 0.819i)10-s + (12.3 − 54.0i)11-s + (25.7 − 12.4i)12-s + (−10.7 + 47.1i)13-s + (−3.69 − 3.52i)14-s + (3.05 + 13.3i)15-s + (38.7 + 48.6i)16-s + (−51.6 + 24.8i)17-s + ⋯
 L(s)  = 1 + (0.0217 − 0.0951i)2-s + (0.433 − 0.543i)3-s + (0.892 + 0.429i)4-s + (−0.211 + 0.265i)5-s + (−0.0422 − 0.0529i)6-s + (0.511 − 0.859i)7-s + (0.121 − 0.151i)8-s + (0.115 + 0.504i)9-s + (0.0206 + 0.0259i)10-s + (0.338 − 1.48i)11-s + (0.619 − 0.298i)12-s + (−0.229 + 1.00i)13-s + (−0.0706 − 0.0673i)14-s + (0.0525 + 0.230i)15-s + (0.605 + 0.759i)16-s + (−0.737 + 0.355i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$49$$    =    $$7^{2}$$ Sign: $0.932 + 0.360i$ Analytic conductor: $$2.89109$$ Root analytic conductor: $$1.70032$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{49} (15, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 49,\ (\ :3/2),\ 0.932 + 0.360i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.76117 - 0.328489i$$ $$L(\frac12)$$ $$\approx$$ $$1.76117 - 0.328489i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1 + (-9.47 + 15.9i)T$$
good2 $$1 + (-0.0613 + 0.268i)T + (-7.20 - 3.47i)T^{2}$$
3 $$1 + (-2.25 + 2.82i)T + (-6.00 - 26.3i)T^{2}$$
5 $$1 + (2.36 - 2.97i)T + (-27.8 - 121. i)T^{2}$$
11 $$1 + (-12.3 + 54.0i)T + (-1.19e3 - 577. i)T^{2}$$
13 $$1 + (10.7 - 47.1i)T + (-1.97e3 - 953. i)T^{2}$$
17 $$1 + (51.6 - 24.8i)T + (3.06e3 - 3.84e3i)T^{2}$$
19 $$1 + 112.T + 6.85e3T^{2}$$
23 $$1 + (108. + 52.3i)T + (7.58e3 + 9.51e3i)T^{2}$$
29 $$1 + (154. - 74.2i)T + (1.52e4 - 1.90e4i)T^{2}$$
31 $$1 + 75.8T + 2.97e4T^{2}$$
37 $$1 + (-358. + 172. i)T + (3.15e4 - 3.96e4i)T^{2}$$
41 $$1 + (53.9 - 67.6i)T + (-1.53e4 - 6.71e4i)T^{2}$$
43 $$1 + (-29.5 - 37.0i)T + (-1.76e4 + 7.75e4i)T^{2}$$
47 $$1 + (-92.7 + 406. i)T + (-9.35e4 - 4.50e4i)T^{2}$$
53 $$1 + (-284. - 136. i)T + (9.28e4 + 1.16e5i)T^{2}$$
59 $$1 + (429. + 539. i)T + (-4.57e4 + 2.00e5i)T^{2}$$
61 $$1 + (221. - 106. i)T + (1.41e5 - 1.77e5i)T^{2}$$
67 $$1 - 367.T + 3.00e5T^{2}$$
71 $$1 + (82.7 + 39.8i)T + (2.23e5 + 2.79e5i)T^{2}$$
73 $$1 + (-12.3 - 53.9i)T + (-3.50e5 + 1.68e5i)T^{2}$$
79 $$1 - 436.T + 4.93e5T^{2}$$
83 $$1 + (-289. - 1.26e3i)T + (-5.15e5 + 2.48e5i)T^{2}$$
89 $$1 + (-17.1 - 75.2i)T + (-6.35e5 + 3.05e5i)T^{2}$$
97 $$1 - 1.26e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$