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Properties

Label 2-1386-77.10-c1-0-36
Degree $2$
Conductor $1386$
Sign $-0.393 + 0.919i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational no
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 + (3.06 − 1.76i)5-s + (0.649 − 2.56i)7-s + 0.999i·8-s + (−1.76 + 3.06i)10-s + (−3.22 + 0.789i)11-s − 5.01·13-s + (0.720 + 2.54i)14-s + (−0.5 − 0.866i)16-s + (1.94 − 3.37i)17-s + (−2.32 − 4.03i)19-s − 3.53i·20-s + (2.39 − 2.29i)22-s + (−0.779 − 1.35i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (1.37 − 0.791i)5-s + (0.245 − 0.969i)7-s + 0.353i·8-s + (−0.559 + 0.969i)10-s + (−0.971 + 0.238i)11-s − 1.39·13-s + (0.192 + 0.680i)14-s + (−0.125 − 0.216i)16-s + (0.472 − 0.817i)17-s + (−0.534 − 0.925i)19-s − 0.791i·20-s + (0.510 − 0.489i)22-s + (−0.162 − 0.281i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.393 + 0.919i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.393 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.055669941\)
\(L(\frac12)\) \(\approx\) \(1.055669941\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.649 + 2.56i)T \)
11 \( 1 + (3.22 - 0.789i)T \)
good5 \( 1 + (-3.06 + 1.76i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 5.01T + 13T^{2} \)
17 \( 1 + (-1.94 + 3.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.32 + 4.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.779 + 1.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.100iT - 29T^{2} \)
31 \( 1 + (0.242 + 0.139i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.352 + 0.610i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.94T + 41T^{2} \)
43 \( 1 - 9.03iT - 43T^{2} \)
47 \( 1 + (5.68 - 3.28i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.77 + 4.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (12.5 + 7.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.56 - 9.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.99 - 3.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.45T + 71T^{2} \)
73 \( 1 + (-1.94 + 3.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.66 + 3.84i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.87T + 83T^{2} \)
89 \( 1 + (-5.15 + 2.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.7iT - 97T^{2} \)
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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

−9.563158531510238706654060382310, −8.548404049835080182490701776016, −7.68190505079472319389408848620, −7.03133327923289630210838991135, −6.07464540218165411727336409946, −4.91597650371287572881453666353, −4.82707829081050065604217065863, −2.77103049243124425800692344067, −1.81587038184515088858139312310, −0.47778953830700785366403127266, 1.89906151630262050684690082498, 2.37084574633511865571521590235, 3.36989450881205382947828561025, 5.05868492169100767836477341486, 5.75018215631482929866437752189, 6.50375258998063472884193543931, 7.56643696054834713344592108499, 8.289221875779737317029141364903, 9.201396180900521812166856582261, 10.00098114472993591733739514693

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