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Properties

Label 2-12e3-216.77-c0-0-1
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  + (0.984 − 0.173i)3-s + (0.939 − 0.342i)9-s + (−0.223 − 1.26i)11-s + (−0.592 − 0.342i)17-s + (0.300 − 0.173i)19-s + (0.939 + 0.342i)25-s + (0.866 − 0.5i)27-s + (−0.439 − 1.20i)33-s + (0.673 + 1.85i)41-s + (−1.85 + 0.326i)43-s + (−0.173 + 0.984i)49-s + (−0.642 − 0.233i)51-s + (0.266 − 0.223i)57-s + (0.342 − 1.93i)59-s + (0.524 + 1.43i)67-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)3-s + (0.939 − 0.342i)9-s + (−0.223 − 1.26i)11-s + (−0.592 − 0.342i)17-s + (0.300 − 0.173i)19-s + (0.939 + 0.342i)25-s + (0.866 − 0.5i)27-s + (−0.439 − 1.20i)33-s + (0.673 + 1.85i)41-s + (−1.85 + 0.326i)43-s + (−0.173 + 0.984i)49-s + (−0.642 − 0.233i)51-s + (0.266 − 0.223i)57-s + (0.342 − 1.93i)59-s + (0.524 + 1.43i)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: $\chi_{1728} (1697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 0.851 + 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.584025688\)
\(L(\frac12)\) \(\approx\) \(1.584025688\)
\(L(1)\) 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 \( 1 + (-0.984 + 0.173i)T \)
good5 \( 1 + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.173 - 0.984i)T^{2} \)
11 \( 1 + (0.223 + 1.26i)T + (-0.939 + 0.342i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.300 + 0.173i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (1.85 - 0.326i)T + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.342 + 1.93i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.524 - 1.43i)T + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (1.62 + 0.592i)T + (0.766 + 0.642i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
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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

−9.392423384806970828623867665557, −8.458314351996783135003712148735, −8.168248498198240385992801368459, −7.07652309136957329423532541381, −6.44562341717404866952200759325, −5.31983338187979192789327951964, −4.35406399074444259601342398719, −3.26919209921236662987521354482, −2.68178954297664127857524099995, −1.26007916128253090733884274516, 1.73366360759929002822850164008, 2.58718999605528195570448522183, 3.69596913541738914915374119361, 4.51049200498972013895269379701, 5.31705514528861807754354902196, 6.70263456261837762171880117314, 7.22238966847900701602572454174, 8.103239831342706950227272595599, 8.797453241370220310591047258095, 9.515616425983794066951372579668

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