↑

Properties

Label 2-1386-7.4-c1-0-24
Degree $2$
Conductor $1386$
Sign $-0.895 + 0.444i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.32 + 2.29i)5-s + (−1.32 − 2.29i)7-s + 0.999·8-s + (1.32 − 2.29i)10-s + (0.5 − 0.866i)11-s − 4·13-s + (−1.32 + 2.29i)14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + (−2.64 − 4.58i)19-s − 2.64·20-s − 0.999·22-s + (−1.32 − 2.29i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.591 + 1.02i)5-s + (−0.499 − 0.866i)7-s + 0.353·8-s + (0.418 − 0.724i)10-s + (0.150 − 0.261i)11-s − 1.10·13-s + (−0.353 + 0.612i)14-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + (−0.606 − 1.05i)19-s − 0.591·20-s − 0.213·22-s + (−0.275 − 0.477i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.895 + 0.444i$
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: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5862501362\)
\(L(\frac12)\) \(\approx\) \(0.5862501362\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 + 2.29i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.32 - 2.29i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.64 + 4.58i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.32 + 2.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.645 + 1.11i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 - 9.29T + 43T^{2} \)
47 \( 1 + (1.96 + 3.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.29 + 5.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.96 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.79 - 6.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.70T + 71T^{2} \)
73 \( 1 + (-7.64 + 13.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.32 + 9.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + (-1.35 - 2.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
show more
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

−9.522463847436696028796787936609, −8.575320013378322844150373481011, −7.53486161170514014936327991057, −6.81101843823685156451680880946, −6.20242970705968244899210772821, −4.81087466797100669891703210818, −3.84829709146850535556966408238, −2.84697860584819490137141063289, −2.04053122196295298169164823317, −0.25638084967829566927432661547, 1.52262873704551230521069236759, 2.64278094051755680939890672805, 4.22533258581579847982253875111, 5.19049410625278915069062530675, 5.70107356161105100327641359652, 6.63440010928272616292494523236, 7.51499310595934183243767400542, 8.501210185984282141751056220491, 9.062084779598781591062718623988, 9.710266239183017629437242877660

Graph of the $Z$-function along the critical line