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Properties

Label 2-1368-1368.563-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.0765 + 0.997i$
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 + 3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)6-s − 0.999·8-s + 9-s + (−0.499 − 0.866i)12-s + (−0.5 + 0.866i)16-s + (1.5 − 0.866i)17-s + (0.5 − 0.866i)18-s − 19-s − 0.999·24-s − 25-s + 27-s + (0.499 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + 3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)6-s − 0.999·8-s + 9-s + (−0.499 − 0.866i)12-s + (−0.5 + 0.866i)16-s + (1.5 − 0.866i)17-s + (0.5 − 0.866i)18-s − 19-s − 0.999·24-s − 25-s + 27-s + (0.499 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0765 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0765 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.0765 + 0.997i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.0765 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.811769925\)
\(L(\frac12)\) \(\approx\) \(1.811769925\)
\(L(1)\) 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 \( 1 - T \)
19 \( 1 + T \)
good5 \( 1 + T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.922741807300913451606231407558, −8.913984797699261443865254747969, −8.179365936341793739816386205435, −7.25719975699544822831324279846, −6.17120516767079900183398245249, −5.14911947558111937717449435008, −4.21864493318544513509819199347, −3.37475428530699469008270890243, −2.54743075305471823987166292303, −1.42634970793287883603469366590, 1.96363466395435767002066044616, 3.31315172622774955439834622362, 3.88253179760994641507756232081, 4.92090122364558420880363377253, 5.92556293128076132485729708758, 6.76580790788425478773297343984, 7.67843693285014608276004957411, 8.207638581912488640544356645592, 8.875759328095496419730436785667, 9.805546413331372149240328574686

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