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Properties

Label 2-1386-33.8-c1-0-4
Degree $2$
Conductor $1386$
Sign $0.121 - 0.992i$
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 + (−0.450 + 0.619i)5-s + (0.951 + 0.309i)7-s + (0.309 + 0.951i)8-s − 0.765i·10-s + (−2.23 − 2.45i)11-s + (2.73 + 3.76i)13-s + (−0.951 + 0.309i)14-s + (−0.809 − 0.587i)16-s + (−0.239 − 0.174i)17-s + (1.24 − 0.403i)19-s + (0.450 + 0.619i)20-s + (3.24 + 0.671i)22-s − 2.28i·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.201 + 0.276i)5-s + (0.359 + 0.116i)7-s + (0.109 + 0.336i)8-s − 0.242i·10-s + (−0.673 − 0.739i)11-s + (0.758 + 1.04i)13-s + (−0.254 + 0.0825i)14-s + (−0.202 − 0.146i)16-s + (−0.0581 − 0.0422i)17-s + (0.284 − 0.0926i)19-s + (0.100 + 0.138i)20-s + (0.692 + 0.143i)22-s − 0.475i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.121 - 0.992i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.121 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.117549415\)
\(L(\frac12)\) \(\approx\) \(1.117549415\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 2.45i)T \)
good5 \( 1 + (0.450 - 0.619i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-2.73 - 3.76i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.239 + 0.174i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.24 + 0.403i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.28iT - 23T^{2} \)
29 \( 1 + (1.67 - 5.16i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.72 + 1.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.636 + 1.95i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.20 - 6.79i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.901iT - 43T^{2} \)
47 \( 1 + (-1.58 + 0.516i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.589 - 0.811i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.26 - 1.38i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.44 - 1.98i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.33T + 67T^{2} \)
71 \( 1 + (7.11 - 9.78i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.0514 + 0.0167i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.07 - 4.23i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.85 - 4.97i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 11.3iT - 89T^{2} \)
97 \( 1 + (-2.24 + 1.63i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.584874597033803107669151949002, −8.839877717642206004503401006045, −8.226995830569582579431070302744, −7.38286931450026387309371141771, −6.60326471170424055697466148602, −5.76358902686703029448866904802, −4.86684232430902521026629713704, −3.70984078137210965897471034270, −2.52765571815495541210961685595, −1.16830675218739514824514306809, 0.64243934779637956698270227472, 1.98369486916961101701479434685, 3.10445431458998412540505453397, 4.16984962527941723376663966217, 5.12205200311543595413318159098, 6.09622476091528954814609266302, 7.26750135168352100732395430880, 7.945583304999031169454688745799, 8.494280756787789232705945198792, 9.450017352825220930203413343531

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