# Properties

 Label 2-7e2-7.2-c3-0-6 Degree $2$ Conductor $49$ Sign $-0.605 + 0.795i$ 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 + (2.5 − 4.33i)2-s + (−8.50 − 14.7i)4-s − 45.0·8-s + (13.5 − 23.3i)9-s + (34 + 58.8i)11-s + (−44.5 + 77.0i)16-s + (−67.5 − 116. i)18-s + 340·22-s + (20 − 34.6i)23-s + (62.5 + 108. i)25-s − 166·29-s + (42.5 + 73.6i)32-s − 459.·36-s + (−225 + 389. i)37-s − 180·43-s + (578. − 1.00e3i)44-s + ⋯
 L(s)  = 1 + (0.883 − 1.53i)2-s + (−1.06 − 1.84i)4-s − 1.98·8-s + (0.5 − 0.866i)9-s + (0.931 + 1.61i)11-s + (−0.695 + 1.20i)16-s + (−0.883 − 1.53i)18-s + 3.29·22-s + (0.181 − 0.314i)23-s + (0.5 + 0.866i)25-s − 1.06·29-s + (0.234 + 0.406i)32-s − 2.12·36-s + (−0.999 + 1.73i)37-s − 0.638·43-s + (1.98 − 3.43i)44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$49$$    =    $$7^{2}$$ Sign: $-0.605 + 0.795i$ Analytic conductor: $$2.89109$$ Root analytic conductor: $$1.70032$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{49} (30, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 49,\ (\ :3/2),\ -0.605 + 0.795i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.884737 - 1.78472i$$ $$L(\frac12)$$ $$\approx$$ $$0.884737 - 1.78472i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1$$
good2 $$1 + (-2.5 + 4.33i)T + (-4 - 6.92i)T^{2}$$
3 $$1 + (-13.5 + 23.3i)T^{2}$$
5 $$1 + (-62.5 - 108. i)T^{2}$$
11 $$1 + (-34 - 58.8i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + 2.19e3T^{2}$$
17 $$1 + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-20 + 34.6i)T + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + 166T + 2.43e4T^{2}$$
31 $$1 + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (225 - 389. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + 6.89e4T^{2}$$
43 $$1 + 180T + 7.95e4T^{2}$$
47 $$1 + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (295 + 510. i)T + (-7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-370 - 640. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 - 688T + 3.57e5T^{2}$$
73 $$1 + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-692 + 1.19e3i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + 5.71e5T^{2}$$
89 $$1 + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$