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Properties

Label 2-1368-152.83-c0-0-0
Degree $2$
Conductor $1368$
Sign $-0.305 - 0.952i$
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 + (−0.499 + 0.866i)4-s − 0.999·8-s + 11-s + (−0.5 − 0.866i)16-s + (1 + 1.73i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)25-s + (0.499 − 0.866i)32-s + (−0.999 + 1.73i)34-s − 0.999·38-s + (−0.5 − 0.866i)41-s + (−1 − 1.73i)43-s + (−0.499 + 0.866i)44-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.999·8-s + 11-s + (−0.5 − 0.866i)16-s + (1 + 1.73i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)25-s + (0.499 − 0.866i)32-s + (−0.999 + 1.73i)34-s − 0.999·38-s + (−0.5 − 0.866i)41-s + (−1 − 1.73i)43-s + (−0.499 + 0.866i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.323359642\)
\(L(\frac12)\) \(\approx\) \(1.323359642\)
\(L(1)\) 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 \( 1 \)
19 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.909641356401712306348470799423, −8.965372255574945273045409780896, −8.310543646827297557422856591370, −7.55921138106756165977651633892, −6.62532180372079981880622406667, −5.95224180534742166519737696206, −5.20185141778349170490553110825, −3.85134957783059151022850704898, −3.62509479241877635208500933010, −1.80031926845812421099671004847, 1.05614377543293206627987499296, 2.45711534118218481231559164559, 3.34667348171865755274524163341, 4.40307498989974494847252125232, 5.09273343030216437140872411861, 6.15383790416726603854635110445, 6.90020675466280367805861618174, 8.053606816114263555851504370460, 9.103891131186112088069976664638, 9.592029729301271249953825001604

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