# Properties

 Label 2-1078-7.2-c1-0-24 Degree $2$ Conductor $1078$ Sign $0.947 + 0.318i$ Analytic cond. $8.60787$ Root an. cond. $2.93391$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 + 0.866i)2-s + (0.707 + 1.22i)3-s + (−0.499 − 0.866i)4-s + (2.12 − 3.67i)5-s − 1.41·6-s + 0.999·8-s + (0.500 − 0.866i)9-s + (2.12 + 3.67i)10-s + (0.5 + 0.866i)11-s + (0.707 − 1.22i)12-s + 6·15-s + (−0.5 + 0.866i)16-s + (−2.82 − 4.89i)17-s + (0.499 + 0.866i)18-s − 4.24·20-s + ⋯
 L(s)  = 1 + (−0.353 + 0.612i)2-s + (0.408 + 0.707i)3-s + (−0.249 − 0.433i)4-s + (0.948 − 1.64i)5-s − 0.577·6-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.670 + 1.16i)10-s + (0.150 + 0.261i)11-s + (0.204 − 0.353i)12-s + 1.54·15-s + (−0.125 + 0.216i)16-s + (−0.685 − 1.18i)17-s + (0.117 + 0.204i)18-s − 0.948·20-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1078$$    =    $$2 \cdot 7^{2} \cdot 11$$ Sign: $0.947 + 0.318i$ Analytic conductor: $$8.60787$$ Root analytic conductor: $$2.93391$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1078} (177, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1078,\ (\ :1/2),\ 0.947 + 0.318i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.733347359$$ $$L(\frac12)$$ $$\approx$$ $$1.733347359$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.5 - 0.866i)T$$
7 $$1$$
11 $$1 + (-0.5 - 0.866i)T$$
good3 $$1 + (-0.707 - 1.22i)T + (-1.5 + 2.59i)T^{2}$$
5 $$1 + (-2.12 + 3.67i)T + (-2.5 - 4.33i)T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 + (2.82 + 4.89i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 - 2T + 29T^{2}$$
31 $$1 + (0.707 + 1.22i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 - 11.3T + 41T^{2}$$
43 $$1 + 8T + 43T^{2}$$
47 $$1 + (-2.12 + 3.67i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (4 + 6.92i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-0.707 - 1.22i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (1.41 - 2.44i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 2T + 71T^{2}$$
73 $$1 + (-4.24 - 7.34i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 16.9T + 83T^{2}$$
89 $$1 + (3.53 - 6.12i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 - 9.89T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$