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Properties

Label 2-12e3-216.149-c0-0-0
Degree $2$
Conductor $1728$
Sign $0.851 + 0.524i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.851 + 0.524i$
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: \((2,\ 1728,\ (\ :0),\ 0.851 + 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9342161723\)
\(L(\frac12)\) \(\approx\) \(0.9342161723\)
\(L(1)\) 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

−9.659617953345784339905465618535, −8.359443347877722930542853394991, −7.78523986060443645774560540254, −7.13173535479334439417886911267, −6.18568881495580892651374769298, −5.47437013714142149739416845653, −4.74076976142338638934557363342, −3.41651496227978527538952999541, −2.29222787398104998103519796997, −1.05996432983675071744677218802, 1.09780757991249140510811360944, 3.01352297380702664036971146317, 3.59754976808402774211395233801, 4.84767230632929533451275520069, 5.49562931204940494862571743476, 6.05336896393545518115652649418, 7.34808845477132884640635570449, 7.85471888203819911876098176218, 9.060364307554250184372851985216, 9.700952093064036125210771423333

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