Properties

Degree $2$
Conductor $1386$
Sign $-0.286 + 0.958i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.96 − 3.40i)5-s + (2.19 + 1.48i)7-s − 0.999i·8-s + (−3.40 + 1.96i)10-s + (−0.866 + 0.5i)11-s − 2.44i·13-s + (−1.15 − 2.38i)14-s + (−0.5 + 0.866i)16-s + (−3.64 − 6.31i)17-s + (3.17 + 1.83i)19-s + 3.93·20-s + 0.999·22-s + (2.94 + 1.69i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.879 − 1.52i)5-s + (0.827 + 0.560i)7-s − 0.353i·8-s + (−1.07 + 0.621i)10-s + (−0.261 + 0.150i)11-s − 0.679i·13-s + (−0.308 − 0.636i)14-s + (−0.125 + 0.216i)16-s + (−0.884 − 1.53i)17-s + (0.729 + 0.421i)19-s + 0.879·20-s + 0.213·22-s + (0.613 + 0.354i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.286 + 0.958i$
Motivic weight: \(1\)
Character: $\chi_{1386} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.286 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.476292129\)
\(L(\frac12)\) \(\approx\) \(1.476292129\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-2.19 - 1.48i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (-1.96 + 3.40i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + (3.64 + 6.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.17 - 1.83i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.94 - 1.69i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.16iT - 29T^{2} \)
31 \( 1 + (1.99 - 1.15i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.98 + 3.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.92T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + (4.31 - 7.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.5 + 6.68i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.975 - 1.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.16 - 2.98i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.33 + 4.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.90iT - 71T^{2} \)
73 \( 1 + (2.37 - 1.37i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.69 + 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.05T + 83T^{2} \)
89 \( 1 + (5.87 - 10.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.85iT - 97T^{2} \)
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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

−9.444622617126758882234767261034, −8.619115119916207267074190836179, −8.062813543921186786974033724269, −7.11977071341331948237143317502, −5.72828238081393421566160233411, −5.20460160483382458466014443209, −4.41616631561134827318673994594, −2.76547544665914397952105995894, −1.83118020492141917830317763902, −0.75844179188059928297878762301, 1.55166826205007776885710591263, 2.44926721889607779474111785565, 3.67889107841272697825666750457, 4.95509800836290745218132061589, 5.94904035415169122778766117024, 6.79425678721530932740993058267, 7.15091630215322444357302019623, 8.207017028094385438909794467940, 8.980595475646738834097004114764, 9.948038082563771770554489723996

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