Properties

Degree $2$
Conductor $1386$
Sign $0.498 + 0.866i$
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 + (−0.346 − 0.199i)5-s + (1.03 − 2.43i)7-s + 0.999·8-s + (0.346 − 0.199i)10-s + (3.01 − 1.38i)11-s − 0.164i·13-s + (1.58 + 2.11i)14-s + (−0.5 + 0.866i)16-s + (0.906 + 1.57i)17-s + (−5.41 − 3.12i)19-s + 0.399i·20-s + (−0.308 + 3.30i)22-s + (3.76 + 2.17i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.154 − 0.0894i)5-s + (0.392 − 0.919i)7-s + 0.353·8-s + (0.109 − 0.0632i)10-s + (0.908 − 0.417i)11-s − 0.0456i·13-s + (0.424 + 0.565i)14-s + (−0.125 + 0.216i)16-s + (0.219 + 0.380i)17-s + (−1.24 − 0.717i)19-s + 0.0894i·20-s + (−0.0656 + 0.704i)22-s + (0.785 + 0.453i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.498 + 0.866i$
Motivic weight: \(1\)
Character: $\chi_{1386} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.498 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.141192757\)
\(L(\frac12)\) \(\approx\) \(1.141192757\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.03 + 2.43i)T \)
11 \( 1 + (-3.01 + 1.38i)T \)
good5 \( 1 + (0.346 + 0.199i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 0.164iT - 13T^{2} \)
17 \( 1 + (-0.906 - 1.57i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.41 + 3.12i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.76 - 2.17i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.12T + 29T^{2} \)
31 \( 1 + (0.141 + 0.245i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.40 + 4.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.23T + 41T^{2} \)
43 \( 1 + 2.07iT - 43T^{2} \)
47 \( 1 + (-0.367 - 0.212i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.71 + 4.45i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.92 + 3.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.10 + 3.52i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0327 - 0.0567i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.43iT - 71T^{2} \)
73 \( 1 + (-7.21 + 4.16i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.531 - 0.306i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.91T + 83T^{2} \)
89 \( 1 + (-8.89 - 5.13i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−9.273399890777351762108391453442, −8.556015334963669333590801588667, −7.86404786687011518396159747932, −6.96266784861469033133089540819, −6.39990267769559002868838560691, −5.32108666179296105488490394953, −4.33960576232032900129806965247, −3.60976700423994372806974448782, −1.86950095173751219641652908194, −0.55179331296738508284855690396, 1.44764323424111812538126488470, 2.40424900975067407701277981612, 3.56174382870260796781411359829, 4.48613371908775662818142436645, 5.48075601715429640113472248179, 6.49811573838720645352694445973, 7.40212111916618030378940183713, 8.368059127847642329301894632427, 8.934425871316848448066161365221, 9.655902318713140582555714856452

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