Properties

Label 2-1368-152.99-c0-0-0
Degree $2$
Conductor $1368$
Sign $0.672 - 0.740i$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.500 + 0.866i)8-s + (0.173 + 0.300i)11-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + (0.766 + 0.642i)19-s + (0.0603 + 0.342i)22-s + (0.173 − 0.984i)25-s + (−0.173 + 0.984i)32-s + (−0.766 − 0.642i)34-s + (0.5 + 0.866i)38-s + (−0.266 − 1.50i)41-s + (−0.766 + 0.642i)43-s + (−0.0603 + 0.342i)44-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.500 + 0.866i)8-s + (0.173 + 0.300i)11-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + (0.766 + 0.642i)19-s + (0.0603 + 0.342i)22-s + (0.173 − 0.984i)25-s + (−0.173 + 0.984i)32-s + (−0.766 − 0.642i)34-s + (0.5 + 0.866i)38-s + (−0.266 − 1.50i)41-s + (−0.766 + 0.642i)43-s + (−0.0603 + 0.342i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.672 - 0.740i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.672 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.904553860\)
\(L(\frac12)\) \(\approx\) \(1.904553860\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 \)
19 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
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 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
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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

−10.00668967021357197950152520171, −8.929333028571894099971059184639, −8.132989832654868891461726244412, −7.24161792724496981537237931069, −6.58995517168230179449110000294, −5.69836785865108672941675352813, −4.82474859965014723984427307759, −4.02334268265414628921121604209, −3.00082475588478984637833464627, −1.89424456873575858825120713011, 1.44625138924848356055359883944, 2.71659461309361275623900224654, 3.59426921854885338489579199497, 4.60591103935087721741175043268, 5.34917796674674607365311047076, 6.32634408214956810603503923595, 6.97522334131577793483354100974, 7.940482120776892392742541174260, 9.058693951261213878468494047946, 9.748575149841909369467442713333

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