Properties

Degree $2$
Conductor $1386$
Sign $0.423 - 0.906i$
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 + (1.87 − 1.07i)5-s + (−2.63 − 0.219i)7-s + 0.999·8-s + (−1.87 − 1.07i)10-s + (3.25 + 0.654i)11-s + 3.53i·13-s + (1.12 + 2.39i)14-s + (−0.5 − 0.866i)16-s + (−3.82 + 6.63i)17-s + (−1.66 + 0.963i)19-s + 2.15i·20-s + (−1.05 − 3.14i)22-s + (−7.21 + 4.16i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.836 − 0.482i)5-s + (−0.996 − 0.0828i)7-s + 0.353·8-s + (−0.591 − 0.341i)10-s + (0.980 + 0.197i)11-s + 0.981i·13-s + (0.301 + 0.639i)14-s + (−0.125 − 0.216i)16-s + (−0.928 + 1.60i)17-s + (−0.383 + 0.221i)19-s + 0.482i·20-s + (−0.225 − 0.670i)22-s + (−1.50 + 0.869i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.423 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.423 - 0.906i$
Motivic weight: \(1\)
Character: $\chi_{1386} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.423 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8780513666\)
\(L(\frac12)\) \(\approx\) \(0.8780513666\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 0.219i)T \)
11 \( 1 + (-3.25 - 0.654i)T \)
good5 \( 1 + (-1.87 + 1.07i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 3.53iT - 13T^{2} \)
17 \( 1 + (3.82 - 6.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.66 - 0.963i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.21 - 4.16i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.79T + 29T^{2} \)
31 \( 1 + (-2.97 + 5.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.55 + 2.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.00T + 41T^{2} \)
43 \( 1 + 2.48iT - 43T^{2} \)
47 \( 1 + (7.34 - 4.23i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.83 - 2.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.95 - 2.28i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.71 - 5.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.46 + 2.53i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.9iT - 71T^{2} \)
73 \( 1 + (0.486 + 0.280i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.63 - 3.82i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.45T + 83T^{2} \)
89 \( 1 + (-0.609 + 0.351i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.23T + 97T^{2} \)
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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

−9.562901660358781945874698672689, −9.239544654720215759583320610325, −8.432343060744043560518023806134, −7.30006832356778785629756958137, −6.25716351247337592049728677844, −5.86205041184777141871782040585, −4.16891345231897262822275894915, −3.87829090954885086584165956070, −2.23492857355100822471307474050, −1.53240772131157663671715043828, 0.38826058096103827334672838030, 2.17649564962908598756402529384, 3.18485819960197710851363946259, 4.43683962082537438920399649299, 5.56309501961402360468093670386, 6.43553244906276658331446101268, 6.65448661329600896065193517167, 7.75334878778314807088098915719, 8.778048461714960876308955675326, 9.417912649119821836138417691393

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