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Properties

Label 2-1368-152.139-c0-0-0
Degree $2$
Conductor $1368$
Sign $0.756 + 0.654i$
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 + (0.173 − 0.984i)4-s + (0.500 + 0.866i)8-s + (−0.939 − 1.62i)11-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + (0.173 − 0.984i)19-s + (1.76 + 0.642i)22-s + (−0.939 + 0.342i)25-s + (0.939 − 0.342i)32-s + (−0.173 + 0.984i)34-s + (0.5 + 0.866i)38-s + (0.326 + 0.118i)41-s + (−0.173 − 0.984i)43-s + (−1.76 + 0.642i)44-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (0.500 + 0.866i)8-s + (−0.939 − 1.62i)11-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + (0.173 − 0.984i)19-s + (1.76 + 0.642i)22-s + (−0.939 + 0.342i)25-s + (0.939 − 0.342i)32-s + (−0.173 + 0.984i)34-s + (0.5 + 0.866i)38-s + (0.326 + 0.118i)41-s + (−0.173 − 0.984i)43-s + (−1.76 + 0.642i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.756 + 0.654i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.756 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6336309026\)
\(L(\frac12)\) \(\approx\) \(0.6336309026\)
\(L(1)\) 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 \( 1 \)
19 \( 1 + (-0.173 + 0.984i)T \)
good5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
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

−9.622808180962944653848812920546, −8.746611717474608195580442197565, −8.124169241453674584131074310009, −7.40922011730144497606656648963, −6.50775291241865228320517542851, −5.55352624545299766623660972295, −5.10322673611250113940444738004, −3.51170385586158802213469212714, −2.38080897339282886204784830074, −0.69525114263984237989861565491, 1.58900805298940183490429102612, 2.52076218624501440137204669672, 3.70676610531026564349526626369, 4.60120072680533270478644518944, 5.76013178781019643266258534869, 6.90377655549106754900771382034, 7.81705843551604311044116741834, 8.075256093340410874528639767127, 9.330412366139883115288004540955, 9.996179495458104161989924920494

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