Properties

Degree $2$
Conductor $1386$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2.89·5-s + 7-s + 8-s − 2.89·10-s + 11-s − 0.364·13-s + 14-s + 16-s − 2.89·17-s + 7.25·19-s − 2.89·20-s + 22-s + 8.14·23-s + 3.36·25-s − 0.364·26-s + 28-s − 2.36·29-s + 10.6·31-s + 32-s − 2.89·34-s − 2.89·35-s − 3.78·37-s + 7.25·38-s − 2.89·40-s + 2.89·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.29·5-s + 0.377·7-s + 0.353·8-s − 0.914·10-s + 0.301·11-s − 0.101·13-s + 0.267·14-s + 0.250·16-s − 0.701·17-s + 1.66·19-s − 0.646·20-s + 0.213·22-s + 1.69·23-s + 0.672·25-s − 0.0715·26-s + 0.188·28-s − 0.439·29-s + 1.91·31-s + 0.176·32-s − 0.496·34-s − 0.488·35-s − 0.622·37-s + 1.17·38-s − 0.457·40-s + 0.451·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1386} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.274720867\)
\(L(\frac12)\) \(\approx\) \(2.274720867\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good5 \( 1 + 2.89T + 5T^{2} \)
13 \( 1 + 0.364T + 13T^{2} \)
17 \( 1 + 2.89T + 17T^{2} \)
19 \( 1 - 7.25T + 19T^{2} \)
23 \( 1 - 8.14T + 23T^{2} \)
29 \( 1 + 2.36T + 29T^{2} \)
31 \( 1 - 10.6T + 31T^{2} \)
37 \( 1 + 3.78T + 37T^{2} \)
41 \( 1 - 2.89T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + 6.52T + 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 + 6.14T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 1.10T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 3.10T + 83T^{2} \)
89 \( 1 - 2.14T + 89T^{2} \)
97 \( 1 - 6.72T + 97T^{2} \)
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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

−9.500381757505576461744869066533, −8.680442375049077547121372089282, −7.68712141052455163926656827676, −7.27101098499960963634692509281, −6.29791579561805203836526474112, −5.12258704213171560060491740868, −4.50583889468581131997651863894, −3.58630503822586028230338006360, −2.72746456133807310065222655570, −1.03809843908470062371467227493, 1.03809843908470062371467227493, 2.72746456133807310065222655570, 3.58630503822586028230338006360, 4.50583889468581131997651863894, 5.12258704213171560060491740868, 6.29791579561805203836526474112, 7.27101098499960963634692509281, 7.68712141052455163926656827676, 8.680442375049077547121372089282, 9.500381757505576461744869066533

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