Properties

Degree $2$
Conductor $1386$
Sign $0.576 - 0.817i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s + 1.06·5-s + (0.884 + 2.49i)7-s + i·8-s − 1.06i·10-s + i·11-s + (2.49 − 0.884i)14-s + 16-s − 3.91·17-s + 7.49i·19-s − 1.06·20-s + 22-s + 5.33i·23-s − 3.85·25-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.478·5-s + (0.334 + 0.942i)7-s + 0.353i·8-s − 0.338i·10-s + 0.301i·11-s + (0.666 − 0.236i)14-s + 0.250·16-s − 0.950·17-s + 1.71i·19-s − 0.239·20-s + 0.213·22-s + 1.11i·23-s − 0.771·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.576 - 0.817i$
Motivic weight: \(1\)
Character: $\chi_{1386} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.576 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.323360758\)
\(L(\frac12)\) \(\approx\) \(1.323360758\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (-0.884 - 2.49i)T \)
11 \( 1 - iT \)
good5 \( 1 - 1.06T + 5T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 3.91T + 17T^{2} \)
19 \( 1 - 7.49iT - 19T^{2} \)
23 \( 1 - 5.33iT - 23T^{2} \)
29 \( 1 + 8.01iT - 29T^{2} \)
31 \( 1 - 3.65iT - 31T^{2} \)
37 \( 1 + 7.33T + 37T^{2} \)
41 \( 1 - 2.50T + 41T^{2} \)
43 \( 1 - 0.230T + 43T^{2} \)
47 \( 1 - 0.370T + 47T^{2} \)
53 \( 1 - 7.56iT - 53T^{2} \)
59 \( 1 - 2.13T + 59T^{2} \)
61 \( 1 - 14.2iT - 61T^{2} \)
67 \( 1 - 2.47T + 67T^{2} \)
71 \( 1 + 3.15iT - 71T^{2} \)
73 \( 1 - 0.370iT - 73T^{2} \)
79 \( 1 + 4.45T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 - 12.8iT - 97T^{2} \)
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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

−9.757097057315385438297205699720, −9.040276395903837191319564266596, −8.284535070344474896527763511591, −7.43514007282953808393834097139, −6.08566402623240550877091369271, −5.59949618541280871314652247888, −4.53520216030556369312090946442, −3.55648454744687246183839416054, −2.31524535992871579662875549010, −1.63989062515641759359024506570, 0.52714240050751429611538598692, 2.12297871926807049308506228298, 3.52432593760422080007073162005, 4.59672317798294441389997814066, 5.18084886733587794468334161416, 6.43255805095231727219498986477, 6.86554754355170227531700767728, 7.74407365933552302117543631410, 8.679931861478424627075176136870, 9.227095633486084599363835726322

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