Properties

 Label 2-1386-7.4-c1-0-22 Degree $2$ Conductor $1386$ Sign $-0.968 - 0.250i$ Analytic cond. $11.0672$ Root an. cond. $3.32675$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $1$

Related objects

Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 + 2.59i)7-s + 0.999·8-s + (−0.5 + 0.866i)11-s − 13-s + (2.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−3 + 5.19i)17-s + (−1 − 1.73i)19-s + 0.999·22-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s + (0.5 + 0.866i)26-s + (−2 − 1.73i)28-s − 9·29-s + ⋯
 L(s)  = 1 + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.188 + 0.981i)7-s + 0.353·8-s + (−0.150 + 0.261i)11-s − 0.277·13-s + (0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.727 + 1.26i)17-s + (−0.229 − 0.397i)19-s + 0.213·22-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s + (0.0980 + 0.169i)26-s + (−0.377 − 0.327i)28-s − 1.67·29-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1386$$    =    $$2 \cdot 3^{2} \cdot 7 \cdot 11$$ Sign: $-0.968 - 0.250i$ Analytic conductor: $$11.0672$$ Root analytic conductor: $$3.32675$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1386} (991, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 1386,\ (\ :1/2),\ -0.968 - 0.250i)$$

Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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$$
3 $$1$$
7 $$1 + (0.5 - 2.59i)T$$
11 $$1 + (0.5 - 0.866i)T$$
good5 $$1 + (-2.5 + 4.33i)T^{2}$$
13 $$1 + T + 13T^{2}$$
17 $$1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + 9T + 29T^{2}$$
31 $$1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 6T + 41T^{2}$$
43 $$1 + 4T + 43T^{2}$$
47 $$1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 6T + 83T^{2}$$
89 $$1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 13T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$