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Properties

Label 2-1368-1368.1139-c0-0-0
Degree $2$
Conductor $1368$
Sign $-0.642 - 0.766i$
Analytic cond. $0.682720$
Root an. cond. $0.826269$
Motivic weight $0$
Arithmetic yes
Rational no
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 + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.5 + 0.866i)11-s + 0.999·12-s + (−0.5 − 0.866i)16-s + 1.73i·17-s − 0.999·18-s + (0.5 + 0.866i)19-s + (−1.5 − 0.866i)22-s + (0.5 + 0.866i)24-s + (0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.5 + 0.866i)11-s + 0.999·12-s + (−0.5 − 0.866i)16-s + 1.73i·17-s − 0.999·18-s + (0.5 + 0.866i)19-s + (−1.5 − 0.866i)22-s + (0.5 + 0.866i)24-s + (0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7801358460\)
\(L(\frac12)\) \(\approx\) \(0.7801358460\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
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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

−10.19992407014213507329314893543, −8.953728124224133310576321510780, −8.002279276761307831718260393146, −7.63670378739673953381509578231, −6.83068697769412381049165946727, −5.86494250506469358928852376920, −5.38045687324001764086945309329, −4.40955263132450233428023203095, −3.14820470194036311902705481718, −1.88196372475311117823275748292, 0.57279851338610339336732845235, 2.73257050027175518378664748740, 3.15647844675266913186141739174, 4.56571333109753388850040806957, 5.04881230319214055912586177595, 5.77333008573524034871407430954, 6.78924731069484555945057345848, 8.110951552174974471932237545516, 9.043044991540750818265817362150, 9.742837198637934637974034440040

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