Properties

Degree $2$
Conductor $1386$
Sign $-0.722 - 0.691i$
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 + (2.42 + 1.39i)5-s + (−2.62 − 0.299i)7-s + 0.999·8-s + (−2.42 + 1.39i)10-s + (−1.36 − 3.02i)11-s + 3.20i·13-s + (1.57 − 2.12i)14-s + (−0.5 + 0.866i)16-s + (2.05 + 3.56i)17-s + (1.82 + 1.05i)19-s − 2.79i·20-s + (3.30 + 0.323i)22-s + (0.902 + 0.521i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (1.08 + 0.625i)5-s + (−0.993 − 0.113i)7-s + 0.353·8-s + (−0.765 + 0.442i)10-s + (−0.413 − 0.910i)11-s + 0.888i·13-s + (0.420 − 0.568i)14-s + (−0.125 + 0.216i)16-s + (0.499 + 0.864i)17-s + (0.419 + 0.242i)19-s − 0.625i·20-s + (0.703 + 0.0690i)22-s + (0.188 + 0.108i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.722 - 0.691i$
Motivic weight: \(1\)
Character: $\chi_{1386} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.101044021\)
\(L(\frac12)\) \(\approx\) \(1.101044021\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.299i)T \)
11 \( 1 + (1.36 + 3.02i)T \)
good5 \( 1 + (-2.42 - 1.39i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 3.20iT - 13T^{2} \)
17 \( 1 + (-2.05 - 3.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.82 - 1.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.902 - 0.521i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.40T + 29T^{2} \)
31 \( 1 + (-3.58 - 6.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.60 - 7.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.110T + 41T^{2} \)
43 \( 1 - 4.20iT - 43T^{2} \)
47 \( 1 + (-7.27 - 4.19i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.3 - 5.96i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.34 + 3.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.21 - 5.32i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.97 + 6.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.0iT - 71T^{2} \)
73 \( 1 + (5.46 - 3.15i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-12.4 - 7.17i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 + (-10.3 - 5.97i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.6T + 97T^{2} \)
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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

−9.757415171944213616315622804631, −9.203814050444602729349290477981, −8.299899652950914295468251514258, −7.31192711771580426152261094042, −6.46118206511515071290156875170, −6.04216151954883648078170586248, −5.22668578129669820775269272328, −3.76064169912820849688782445614, −2.82103737879341456974249106350, −1.47158149922153288533982923114, 0.50268999027402944495148341564, 1.97421598926858218753409190885, 2.82404465499193639111778321991, 3.94078811748521276351257689080, 5.29674099200803428498792231020, 5.64051349652807696213989940232, 6.98322648319030368270412054610, 7.66724364048744523786826083128, 8.821300984322638352732619139777, 9.534395238017949109158180858801

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