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Properties

Label 2-1368-1368.1283-c0-0-1
Degree $2$
Conductor $1368$
Sign $-0.479 + 0.877i$
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  + (0.766 − 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)6-s + (−0.500 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (0.592 + 0.342i)11-s + (−0.766 − 0.642i)12-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)18-s + (−0.173 + 0.984i)19-s + (0.673 − 0.118i)22-s − 24-s + (−0.173 + 0.984i)25-s − 0.999·27-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)6-s + (−0.500 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (0.592 + 0.342i)11-s + (−0.766 − 0.642i)12-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)18-s + (−0.173 + 0.984i)19-s + (0.673 − 0.118i)22-s − 24-s + (−0.173 + 0.984i)25-s − 0.999·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.479 + 0.877i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.479 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.857463218\)
\(L(\frac12)\) \(\approx\) \(1.857463218\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
good5 \( 1 + (0.173 - 0.984i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.592 - 0.342i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.826 - 0.984i)T + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (0.766 + 0.642i)T^{2} \)
83 \( 1 - 0.684iT - T^{2} \)
89 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \)
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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

−9.496307841621629151118853231757, −8.886632476629208105608535642718, −7.77585403605047612177700781380, −6.96955035247569827231864603672, −6.17857758781170475241036887365, −5.38199518138316089213686758654, −4.10935509045699686433266793818, −3.38131907039227865141562846544, −2.24604976495019935367240968894, −1.33726736912528547769660902196, 2.37207282571923225113383502385, 3.30789245165704728807045291696, 4.20090137508220199579503228898, 4.86582021331510076878676020842, 5.83519081347908445780983438988, 6.67281083819649461378571692926, 7.63315530025020252745523201261, 8.493803594979844857981582864546, 9.009385632166159468227595125487, 9.936570914281488007843794489556

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