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Properties

Label 2-12e3-216.149-c0-0-1
Degree $2$
Conductor $1728$
Sign $0.524 - 0.851i$
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  + (0.642 + 0.766i)3-s + (−0.173 + 0.984i)9-s + (0.524 − 0.439i)11-s + (1.70 + 0.984i)17-s + (−1.32 + 0.766i)19-s + (−0.173 − 0.984i)25-s + (−0.866 + 0.500i)27-s + (0.673 + 0.118i)33-s + (1.26 + 0.223i)41-s + (0.223 + 0.266i)43-s + (−0.766 − 0.642i)49-s + (0.342 + 1.93i)51-s + (−1.43 − 0.524i)57-s + (0.984 + 0.826i)59-s + (−1.85 − 0.326i)67-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)3-s + (−0.173 + 0.984i)9-s + (0.524 − 0.439i)11-s + (1.70 + 0.984i)17-s + (−1.32 + 0.766i)19-s + (−0.173 − 0.984i)25-s + (−0.866 + 0.500i)27-s + (0.673 + 0.118i)33-s + (1.26 + 0.223i)41-s + (0.223 + 0.266i)43-s + (−0.766 − 0.642i)49-s + (0.342 + 1.93i)51-s + (−1.43 − 0.524i)57-s + (0.984 + 0.826i)59-s + (−1.85 − 0.326i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.524 - 0.851i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 0.524 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.431389117\)
\(L(\frac12)\) \(\approx\) \(1.431389117\)
\(L(1)\) 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 \( 1 + (-0.642 - 0.766i)T \)
good5 \( 1 + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.524 + 0.439i)T + (0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-1.70 - 0.984i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.32 - 0.766i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.26 - 0.223i)T + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.223 - 0.266i)T + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.984 - 0.826i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (1.85 + 0.326i)T + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.300 + 1.70i)T + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
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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

−9.731850186834608609525283991710, −8.742147619838937066992244951794, −8.245855980504436023488328171028, −7.54012641307481176310182504114, −6.22447884678874661299875171559, −5.66990910050543165968201883013, −4.42304988613756041416467245716, −3.82314432191690581344671909579, −2.91038200973107212401893996476, −1.66702482944725020392179195104, 1.17056272681199540859902103878, 2.35881529164247307111705475122, 3.28562543631308725523060099537, 4.26757656935865196033526406812, 5.41761277491083940750928498418, 6.33822972631082758716890936656, 7.17905323628529530361231066997, 7.67206466056263924884794365459, 8.599060599155582273141270236579, 9.322730816874255052951062931267

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