Properties

Label 2-12e3-36.31-c0-0-1
Degree $2$
Conductor $1728$
Sign $0.642 + 0.766i$
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  + (0.5 − 0.866i)5-s + (0.866 − 0.5i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s − 0.999i·35-s + (0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.5i)47-s + 0.999i·55-s + (0.866 + 0.5i)59-s + (0.5 + 0.866i)61-s + (−0.499 − 0.866i)65-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (0.866 − 0.5i)7-s + (−0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s − 0.999i·35-s + (0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.5i)47-s + 0.999i·55-s + (0.866 + 0.5i)59-s + (0.5 + 0.866i)61-s + (−0.499 − 0.866i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.308050187\)
\(L(\frac12)\) \(\approx\) \(1.308050187\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.390937886570871985252176906990, −8.579167839866973821029247455050, −7.82520614483452988238121293680, −7.27583185136222156985575791669, −5.94774051497234796275808604794, −5.24248666134070464560400147146, −4.66259989987023571304349153822, −3.53293102406919621341779264794, −2.20187990991086008338375275662, −1.12131364256116508425898221565, 1.70005822714374503798060240629, 2.62627162503586019791877367352, 3.59078343140711757516828772445, 4.90457158059993175440515498001, 5.49776867271052208112105938889, 6.52232047004454338084035725251, 7.09118454118665959403419269479, 8.218622599647811835367795538002, 8.698173160389272440767811078940, 9.642000638939927148493646718178

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