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Properties

Label 2-1368-1368.427-c0-0-0
Degree $2$
Conductor $1368$
Sign $0.513 + 0.858i$
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 + (0.939 − 1.62i)11-s + (0.766 + 0.642i)12-s + (−0.939 − 0.342i)16-s + (0.347 + 1.96i)17-s + (−0.939 − 0.342i)18-s + (0.173 − 0.984i)19-s + (−0.326 − 1.85i)22-s + 24-s + (0.173 − 0.984i)25-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 + (0.939 − 1.62i)11-s + (0.766 + 0.642i)12-s + (−0.939 − 0.342i)16-s + (0.347 + 1.96i)17-s + (−0.939 − 0.342i)18-s + (0.173 − 0.984i)19-s + (−0.326 − 1.85i)22-s + 24-s + (0.173 − 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.513 + 0.858i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.513 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.397999386\)
\(L(\frac12)\) \(\approx\) \(1.397999386\)
\(L(1)\) 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 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \)
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 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
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 \( 1 + 1.87T + T^{2} \)
89 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
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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

−9.975601218546094935432689997476, −8.958872784675872317364721733578, −8.430387217120391586080036233494, −6.69817806123584742067373611220, −6.06577251986497808356074229636, −5.48508139962865620292755556074, −4.30920025218086705909136121772, −3.76051461722892091001842965484, −2.84304273566603250719365898479, −1.09936068317688883055780696381, 1.70564110253945768259922854007, 2.88649270189946465254711362946, 4.18238351842750039770590518655, 5.06388924462524712183139555417, 5.75202385904808468400502926948, 6.84963724083686641128423649060, 7.19925905083475957853939372135, 7.83693846475289385232569228287, 9.019146231909413812430179852405, 9.767552092469209995485604433067

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