# Properties

 Degree $2$ Conductor $1386$ Sign $-0.0523 + 0.998i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s − 4-s + 0.929·5-s + (−2.07 − 1.63i)7-s − i·8-s + 0.929i·10-s − i·11-s + (1.63 − 2.07i)14-s + 16-s − 2.34·17-s + 6.87i·19-s − 0.929·20-s + 22-s − 3.04i·23-s − 4.13·25-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.5·4-s + 0.415·5-s + (−0.785 − 0.619i)7-s − 0.353i·8-s + 0.294i·10-s − 0.301i·11-s + (0.437 − 0.555i)14-s + 0.250·16-s − 0.569·17-s + 1.57i·19-s − 0.207·20-s + 0.213·22-s − 0.635i·23-s − 0.827·25-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0523 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0523 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1386$$    =    $$2 \cdot 3^{2} \cdot 7 \cdot 11$$ Sign: $-0.0523 + 0.998i$ Motivic weight: $$1$$ Character: $\chi_{1386} (881, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1386,\ (\ :1/2),\ -0.0523 + 0.998i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6149948469$$ $$L(\frac12)$$ $$\approx$$ $$0.6149948469$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
3 $$1$$
7 $$1 + (2.07 + 1.63i)T$$
11 $$1 + iT$$
good5 $$1 - 0.929T + 5T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + 2.34T + 17T^{2}$$
19 $$1 - 6.87iT - 19T^{2}$$
23 $$1 + 3.04iT - 23T^{2}$$
29 $$1 + 6.39iT - 29T^{2}$$
31 $$1 + 10.0iT - 31T^{2}$$
37 $$1 + 5.04T + 37T^{2}$$
41 $$1 + 10.1T + 41T^{2}$$
43 $$1 - 6.15T + 43T^{2}$$
47 $$1 + 12.0T + 47T^{2}$$
53 $$1 + 11.2iT - 53T^{2}$$
59 $$1 - 1.85T + 59T^{2}$$
61 $$1 + 10.2iT - 61T^{2}$$
67 $$1 + 0.0878T + 67T^{2}$$
71 $$1 + 11.5iT - 71T^{2}$$
73 $$1 - 12.0iT - 73T^{2}$$
79 $$1 - 13.5T + 79T^{2}$$
83 $$1 - 4.06T + 83T^{2}$$
89 $$1 + 17.0T + 89T^{2}$$
97 $$1 - 15.6iT - 97T^{2}$$
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.616692430225485365789504549196, −8.310216966442115966974285251321, −7.85320340598134582747319891128, −6.69498838651167334543651160987, −6.23932144138914177020585190134, −5.40687299885042985088687147085, −4.19992945404993743800534825904, −3.49766467689644574256715391732, −2.00412784936308371621538383241, −0.23962020225016066391790956021, 1.60222225534461171303534266997, 2.69670092259974169606776059556, 3.47118956810352184933443199300, 4.76350709135407470611765993586, 5.45388163549011253809594005566, 6.55061747485704364528393954632, 7.20594376333783704073141459509, 8.621507490930312411826907581309, 9.040050939115512683272694284726, 9.790202759411511018132792599972