Properties

Degree $2$
Conductor $1386$
Sign $0.143 + 0.989i$
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 + (−2.19 − 2.48i)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 + (−0.661 − 0.750i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.143 + 0.989i$
Motivic weight: \(1\)
Character: $\chi_{1386} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.143 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.062203425\)
\(L(\frac12)\) \(\approx\) \(1.062203425\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 2.48i)T \)
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

−9.328982993986447146090339294947, −8.440100714811513150349859059246, −7.988183802868940294774486959145, −7.29500352769240861580412218641, −6.05033037617355929540911296590, −5.04823612123766121222355121699, −4.06112507143843961645939897054, −3.17609325252395996489713323363, −2.15318236805746169664584590884, −0.59720886297965329407934097803, 1.08606568646822194805977309932, 2.48227954177480774909268171410, 4.12551889162415478652815806557, 4.80960680975070282564133638164, 5.37450738994756373335334965987, 6.89751392940451285008810176708, 7.36511114535891890935710341901, 8.005436383794280027839624405170, 9.003608367872292960692609857576, 9.350362631246564206499642090480

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