Properties

Degree $2$
Conductor $1386$
Sign $0.958 + 0.286i$
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 + (1.25 − 2.18i)5-s + (1.48 − 2.19i)7-s + 0.999i·8-s + (2.18 − 1.25i)10-s + (0.866 − 0.5i)11-s + 2.44i·13-s + (2.38 − 1.15i)14-s + (−0.5 + 0.866i)16-s + (2.64 + 4.58i)17-s + (−4.52 − 2.61i)19-s + 2.51·20-s + 0.999·22-s + (4.28 + 2.47i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.562 − 0.975i)5-s + (0.560 − 0.827i)7-s + 0.353i·8-s + (0.689 − 0.398i)10-s + (0.261 − 0.150i)11-s + 0.679i·13-s + (0.636 − 0.308i)14-s + (−0.125 + 0.216i)16-s + (0.641 + 1.11i)17-s + (−1.03 − 0.599i)19-s + 0.562·20-s + 0.213·22-s + (0.893 + 0.515i)23-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.892509079\)
\(L(\frac12)\) \(\approx\) \(2.892509079\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 2.19i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-1.25 + 2.18i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (-2.64 - 4.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.52 + 2.61i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.28 - 2.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.05iT - 29T^{2} \)
31 \( 1 + (-7.44 + 4.29i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.91 + 10.2i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.92T + 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 + (-0.637 + 1.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.414 + 0.239i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.02 - 5.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.7 - 6.20i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.11 + 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.7iT - 71T^{2} \)
73 \( 1 + (8.52 - 4.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.52 + 6.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.35T + 83T^{2} \)
89 \( 1 + (7.92 - 13.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.6iT - 97T^{2} \)
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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

−9.465157574766825197392798348157, −8.571651471568848898324705668120, −7.979838240517680436304877210941, −6.96750353803004794810625265312, −6.15489172999411889202410266733, −5.31526877318079307804579840869, −4.41649248037286536983110249119, −3.88757481733624123063091302289, −2.26017035062498667562823716689, −1.10914939931494547880564605440, 1.48506277649331084175485344498, 2.71316496552743646045531460394, 3.16943071744283706932018298869, 4.72596176044412169826911629634, 5.28898900468590736055801560862, 6.36355799848002936525711453691, 6.81780016891302037950354870130, 8.047931213812974532380872537534, 8.824487882167119639063222557267, 9.953396173781356624472883500722

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