Properties

Degree $2$
Conductor $1386$
Sign $-0.347 + 0.937i$
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 + (−1.78 − 1.02i)5-s + (0.289 − 2.62i)7-s + 0.999·8-s + (1.78 − 1.02i)10-s + (1.29 − 3.05i)11-s + 6.97i·13-s + (2.13 + 1.56i)14-s + (−0.5 + 0.866i)16-s + (−2.66 − 4.61i)17-s + (7.36 + 4.25i)19-s + 2.05i·20-s + (1.99 + 2.64i)22-s + (−1.35 − 0.782i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.797 − 0.460i)5-s + (0.109 − 0.993i)7-s + 0.353·8-s + (0.563 − 0.325i)10-s + (0.390 − 0.920i)11-s + 1.93i·13-s + (0.569 + 0.418i)14-s + (−0.125 + 0.216i)16-s + (−0.646 − 1.11i)17-s + (1.69 + 0.975i)19-s + 0.460i·20-s + (0.425 + 0.564i)22-s + (−0.282 − 0.163i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.347 + 0.937i$
Motivic weight: \(1\)
Character: $\chi_{1386} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.347 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6358795645\)
\(L(\frac12)\) \(\approx\) \(0.6358795645\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.289 + 2.62i)T \)
11 \( 1 + (-1.29 + 3.05i)T \)
good5 \( 1 + (1.78 + 1.02i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 6.97iT - 13T^{2} \)
17 \( 1 + (2.66 + 4.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.36 - 4.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.35 + 0.782i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.87T + 29T^{2} \)
31 \( 1 + (4.39 + 7.60i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.82 - 3.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.42T + 41T^{2} \)
43 \( 1 + 7.55iT - 43T^{2} \)
47 \( 1 + (10.2 + 5.91i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.93 - 5.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.49 + 2.01i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.39 + 2.53i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.75 + 8.23i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.30iT - 71T^{2} \)
73 \( 1 + (0.828 - 0.478i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.75 - 4.47i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.50T + 83T^{2} \)
89 \( 1 + (-2.99 - 1.72i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.95T + 97T^{2} \)
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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

−9.301729175083457526291096738868, −8.413824985894751791988717088352, −7.70134123842224370653903092899, −6.99797386215829484605497056307, −6.27648045965203895940735952601, −5.04263315167911549502420947791, −4.26106757730394291278959841287, −3.51060627612020860381084585029, −1.59324272918527313543605207077, −0.30912685784572215564667972847, 1.50630574278993880721126828877, 2.88538967555270951433430996636, 3.41637906156816638533760735898, 4.71895594739827292672564744357, 5.53142164760143991884291501104, 6.72973389427531600289724734814, 7.65006002688825642401521735091, 8.192804644815906517613774376695, 9.094337196152137048691312025536, 9.838269798249630487276394529958

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