Properties

Degree $2$
Conductor $1386$
Sign $-0.917 + 0.398i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2.34 + 1.35i)5-s + (0.222 − 2.63i)7-s + 0.999·8-s + (2.34 + 1.35i)10-s + (2.77 + 1.82i)11-s − 2.18i·13-s + (−2.39 + 1.12i)14-s + (−0.5 − 0.866i)16-s + (0.0281 − 0.0488i)17-s + (−2.38 + 1.37i)19-s − 2.71i·20-s + (0.192 − 3.31i)22-s + (−6.02 + 3.47i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−1.05 + 0.606i)5-s + (0.0839 − 0.996i)7-s + 0.353·8-s + (0.742 + 0.428i)10-s + (0.835 + 0.549i)11-s − 0.604i·13-s + (−0.639 + 0.300i)14-s + (−0.125 − 0.216i)16-s + (0.00683 − 0.0118i)17-s + (−0.547 + 0.315i)19-s − 0.606i·20-s + (0.0409 − 0.705i)22-s + (−1.25 + 0.725i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.917 + 0.398i$
Motivic weight: \(1\)
Character: $\chi_{1386} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.917 + 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5263564118\)
\(L(\frac12)\) \(\approx\) \(0.5263564118\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.222 + 2.63i)T \)
11 \( 1 + (-2.77 - 1.82i)T \)
good5 \( 1 + (2.34 - 1.35i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 2.18iT - 13T^{2} \)
17 \( 1 + (-0.0281 + 0.0488i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.38 - 1.37i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.02 - 3.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.43T + 29T^{2} \)
31 \( 1 + (-3.65 + 6.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.62 + 2.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.71T + 41T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 + (-2.58 + 1.49i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.1 + 5.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (12.2 + 7.07i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.2 - 5.93i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.749 + 1.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.82iT - 71T^{2} \)
73 \( 1 + (6.82 + 3.93i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.82 + 4.51i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.2T + 83T^{2} \)
89 \( 1 + (7.43 - 4.29i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.25T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.409150405569946229813038510047, −8.261415979285168475919430430261, −7.70958269377622754303772893372, −7.05645676487302710761813141766, −6.08646237265590160320010925182, −4.51023407454760137802464065162, −3.95091344116301849388570687790, −3.16992771943205116644169865155, −1.74312255308013831794577122425, −0.26223338974166553280824999508, 1.34197419229890850482355382810, 2.90954754148063250421298624502, 4.29654997395498638286514104864, 4.73890185363268180529499445128, 6.11567172016189978933988502306, 6.48676162502249516213966342128, 7.74126122834955782724530533075, 8.419759349022916928891575993071, 8.820238677770025770253526374176, 9.605541454943964416841552305710

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