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Properties

Label 2-12e3-216.173-c0-0-0
Degree $2$
Conductor $1728$
Sign $0.474 - 0.880i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
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  + (0.342 + 0.939i)3-s + (−0.766 + 0.642i)9-s + (1.85 − 0.673i)11-s + (−1.11 + 0.642i)17-s + (1.62 + 0.939i)19-s + (−0.766 − 0.642i)25-s + (−0.866 − 0.500i)27-s + (1.26 + 1.50i)33-s + (−0.439 − 0.524i)41-s + (0.524 + 1.43i)43-s + (0.939 + 0.342i)49-s + (−0.984 − 0.826i)51-s + (−0.326 + 1.85i)57-s + (−0.642 − 0.233i)59-s + (−0.223 − 0.266i)67-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)3-s + (−0.766 + 0.642i)9-s + (1.85 − 0.673i)11-s + (−1.11 + 0.642i)17-s + (1.62 + 0.939i)19-s + (−0.766 − 0.642i)25-s + (−0.866 − 0.500i)27-s + (1.26 + 1.50i)33-s + (−0.439 − 0.524i)41-s + (0.524 + 1.43i)43-s + (0.939 + 0.342i)49-s + (−0.984 − 0.826i)51-s + (−0.326 + 1.85i)57-s + (−0.642 − 0.233i)59-s + (−0.223 − 0.266i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.474 - 0.880i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 0.474 - 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.339175586\)
\(L(\frac12)\) \(\approx\) \(1.339175586\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.342 - 0.939i)T \)
good5 \( 1 + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.939 - 0.342i)T^{2} \)
11 \( 1 + (-1.85 + 0.673i)T + (0.766 - 0.642i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (1.11 - 0.642i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.62 - 0.939i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.939 + 0.342i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.439 + 0.524i)T + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.524 - 1.43i)T + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.642 + 0.233i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.223 + 0.266i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (1.32 + 1.11i)T + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.597079444843441193967750860656, −8.923125624072662080826898523031, −8.342662739595662086726087150078, −7.35226864471575931880323864837, −6.23374220898641025239040487440, −5.69154463340993258629623479307, −4.40969722503602119373553718177, −3.88136391961597574254918098915, −3.00369565539197672550373788351, −1.56983554518555460677441028737, 1.16683846686357089466109265926, 2.21339258715872389024433624788, 3.33690739771893646428898393111, 4.29566669617892149184941725422, 5.41680569391951335865612430044, 6.43447810190844890539576031767, 7.09372040002928131292935370751, 7.50666755634522102960401086765, 8.834013029197810575265156263975, 9.131075342219141962442914433931

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