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Properties

Label 2-1368-1368.203-c0-0-0
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (0.5 + 0.866i)6-s + (0.500 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−1.11 − 0.642i)11-s + (0.173 + 0.984i)12-s + (0.173 + 0.984i)16-s + (−1.11 − 1.32i)17-s + (−0.173 + 0.984i)18-s + (0.939 − 0.342i)19-s + (−0.826 − 0.984i)22-s + (−0.173 + 0.984i)24-s + (−0.766 − 0.642i)25-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (0.5 + 0.866i)6-s + (0.500 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−1.11 − 0.642i)11-s + (0.173 + 0.984i)12-s + (0.173 + 0.984i)16-s + (−1.11 − 1.32i)17-s + (−0.173 + 0.984i)18-s + (0.939 − 0.342i)19-s + (−0.826 − 0.984i)22-s + (−0.173 + 0.984i)24-s + (−0.766 − 0.642i)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: $\chi_{1368} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.266441361\)
\(L(\frac12)\) \(\approx\) \(2.266441361\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
good5 \( 1 + (0.766 + 0.642i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (1.53 - 1.28i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T^{2} \)
83 \( 1 - 1.96iT - T^{2} \)
89 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.233 - 0.642i)T + (-0.766 - 0.642i)T^{2} \)
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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

−9.898231853544297441894470919630, −9.067019743168403584107337296861, −8.147534188843261714723839400920, −7.59203061852223584614872778299, −6.66725023703850083978573582427, −5.46883645571485802778459051505, −4.91755769858708362652549847344, −3.98994365750408857189329639029, −2.95688091003563681689845702009, −2.36391563597029031925050822851, 1.64556193785693756510503380189, 2.45630850573592647540502660675, 3.47914101390827961938291279364, 4.34630680043985530844263352463, 5.45218920283247206254604948023, 6.27937115662792524072375076996, 7.21688692488088198909676197938, 7.79001368859259468579815839426, 8.775041868948902643357894136783, 9.801913005650415934876275293749

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