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Properties

Label 2-1368-1368.227-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.342 + 0.939i$
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 + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s − 0.999·12-s + (−0.5 + 0.866i)16-s − 1.73i·17-s + 0.999·18-s + (0.5 + 0.866i)19-s + (1.5 − 0.866i)22-s + (0.5 − 0.866i)24-s + (0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s − 0.999·12-s + (−0.5 + 0.866i)16-s − 1.73i·17-s + 0.999·18-s + (0.5 + 0.866i)19-s + (1.5 − 0.866i)22-s + (0.5 − 0.866i)24-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7576253594\)
\(L(\frac12)\) \(\approx\) \(0.7576253594\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.432458904204867920238308335262, −8.501454832972385150934496193076, −8.019836385334356519837661368836, −7.33176718438473928179658605471, −6.56436377824123835543205572144, −5.62438940951620278398759517304, −4.95198298515001105788472952546, −3.34901260044235967539600621043, −2.29550226408341643665090498388, −0.69422440525209634359608093189, 1.89987665586571166403869165124, 2.84177364312306092326679357149, 3.71961002244261378867172309241, 4.71907367674055481386319982272, 5.36404037193154562530180335149, 6.99268630818163079759017963548, 7.978537459047402292229944809644, 8.379189087072538668641639368309, 9.395154639966438386803566134467, 9.914239862927693526390715025884

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