Properties

Label 2-12e3-4.3-c2-0-27
Degree $2$
Conductor $1728$
Sign $1$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.76·5-s − 12.0i·7-s + 13.3i·11-s − 8.15·13-s − 4.15·17-s + 8.87i·19-s + 33.5i·23-s − 21.8·25-s + 36.4·29-s − 0.590i·31-s + 21.1i·35-s + 69.8·37-s + 57.5·41-s + 23.5i·43-s − 24.4i·47-s + ⋯
L(s)  = 1  − 0.352·5-s − 1.71i·7-s + 1.21i·11-s − 0.627·13-s − 0.244·17-s + 0.466i·19-s + 1.46i·23-s − 0.875·25-s + 1.25·29-s − 0.0190i·31-s + 0.605i·35-s + 1.88·37-s + 1.40·41-s + 0.548i·43-s − 0.520i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.595896152\)
\(L(\frac12)\) \(\approx\) \(1.595896152\)
\(L(2)\) 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 \( 1 + 1.76T + 25T^{2} \)
7 \( 1 + 12.0iT - 49T^{2} \)
11 \( 1 - 13.3iT - 121T^{2} \)
13 \( 1 + 8.15T + 169T^{2} \)
17 \( 1 + 4.15T + 289T^{2} \)
19 \( 1 - 8.87iT - 361T^{2} \)
23 \( 1 - 33.5iT - 529T^{2} \)
29 \( 1 - 36.4T + 841T^{2} \)
31 \( 1 + 0.590iT - 961T^{2} \)
37 \( 1 - 69.8T + 1.36e3T^{2} \)
41 \( 1 - 57.5T + 1.68e3T^{2} \)
43 \( 1 - 23.5iT - 1.84e3T^{2} \)
47 \( 1 + 24.4iT - 2.20e3T^{2} \)
53 \( 1 + 29.9T + 2.80e3T^{2} \)
59 \( 1 + 55.0iT - 3.48e3T^{2} \)
61 \( 1 - 11.1T + 3.72e3T^{2} \)
67 \( 1 - 99.6iT - 4.48e3T^{2} \)
71 \( 1 + 71.2iT - 5.04e3T^{2} \)
73 \( 1 - 36.0T + 5.32e3T^{2} \)
79 \( 1 + 108. iT - 6.24e3T^{2} \)
83 \( 1 + 100. iT - 6.88e3T^{2} \)
89 \( 1 - 159.T + 7.92e3T^{2} \)
97 \( 1 + 24.0T + 9.40e3T^{2} \)
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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

−9.445438232741876346840202296596, −7.961461232390973091191334958452, −7.59559112384997744004638466358, −6.98172806672931156734476089870, −6.01216311380060878507860659158, −4.67410202199876435583687115907, −4.28591753979879512379538318413, −3.32110113727956984480117489679, −1.94618432263626450164782098846, −0.75387974284377392798530027722, 0.62856134703068543009087940889, 2.40590344529190265202791950219, 2.83893795070776727102242352719, 4.19826486937830891086079847244, 5.10374034119910242843435628437, 5.98950894228802114698880583262, 6.48875256357166766918069402011, 7.80534191994130560945030037245, 8.389877406433222076920258698225, 9.045644695604444412363442046050

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