Properties

Label 2-1386-33.8-c1-0-10
Degree $2$
Conductor $1386$
Sign $0.967 + 0.251i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−1.25 + 1.72i)5-s + (−0.951 − 0.309i)7-s + (−0.309 − 0.951i)8-s + 2.12i·10-s + (2.72 − 1.88i)11-s + (0.138 + 0.190i)13-s + (−0.951 + 0.309i)14-s + (−0.809 − 0.587i)16-s + (3.83 + 2.78i)17-s + (2.37 − 0.770i)19-s + (1.25 + 1.72i)20-s + (1.09 − 3.13i)22-s + 7.44i·23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.559 + 0.770i)5-s + (−0.359 − 0.116i)7-s + (−0.109 − 0.336i)8-s + 0.673i·10-s + (0.821 − 0.569i)11-s + (0.0383 + 0.0527i)13-s + (−0.254 + 0.0825i)14-s + (−0.202 − 0.146i)16-s + (0.931 + 0.676i)17-s + (0.543 − 0.176i)19-s + (0.279 + 0.385i)20-s + (0.233 − 0.667i)22-s + 1.55i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.967 + 0.251i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.967 + 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.165783786\)
\(L(\frac12)\) \(\approx\) \(2.165783786\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-2.72 + 1.88i)T \)
good5 \( 1 + (1.25 - 1.72i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-0.138 - 0.190i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.83 - 2.78i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.37 + 0.770i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 7.44iT - 23T^{2} \)
29 \( 1 + (-1.54 + 4.75i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.902 + 0.656i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.09 + 9.53i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.856 - 2.63i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.37iT - 43T^{2} \)
47 \( 1 + (-11.7 + 3.80i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.73 - 6.51i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-12.9 - 4.20i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.05 - 9.70i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.74T + 67T^{2} \)
71 \( 1 + (-5.46 + 7.51i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.11 - 1.33i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.17 + 5.74i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.89 - 1.37i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.21iT - 89T^{2} \)
97 \( 1 + (-0.932 + 0.677i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.665896439184575780076903429097, −8.903521024344801978480198628013, −7.66972783772814902977964510743, −7.14012915993509430050260858415, −6.06714533566656156685444529885, −5.51173638708618309340288790556, −3.96455696252981410010105441499, −3.64898124632479933279257577815, −2.61062251571655182886722384748, −1.08362440693107715420591191493, 0.982440553752426554061191931509, 2.68422773328604600140942944836, 3.75421105632977921922628166636, 4.59467833324548923582685444910, 5.27868312309239675346140654983, 6.38706589695434599988773343772, 7.04489718490125524588713330845, 7.990718295432839877497009271750, 8.654612579249927802224881015900, 9.492719452553635603068999094690

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