Properties

Label 2-12e3-4.3-c2-0-38
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  + 5.13·5-s − 4.02i·7-s − 4.30i·11-s + 18.4·13-s + 23.5·17-s + 21.7i·19-s + 30.7i·23-s + 1.34·25-s − 12.6·29-s + 24.5i·31-s − 20.6i·35-s − 18.2·37-s + 38.0·41-s − 34.9i·43-s − 29.6i·47-s + ⋯
L(s)  = 1  + 1.02·5-s − 0.575i·7-s − 0.391i·11-s + 1.42·13-s + 1.38·17-s + 1.14i·19-s + 1.33i·23-s + 0.0536·25-s − 0.437·29-s + 0.791i·31-s − 0.590i·35-s − 0.492·37-s + 0.929·41-s − 0.813i·43-s − 0.630i·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

\(L(\frac{3}{2})\) \(\approx\) \(2.860603284\)
\(L(\frac12)\) \(\approx\) \(2.860603284\)
\(L(2)\) 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 \( 1 - 5.13T + 25T^{2} \)
7 \( 1 + 4.02iT - 49T^{2} \)
11 \( 1 + 4.30iT - 121T^{2} \)
13 \( 1 - 18.4T + 169T^{2} \)
17 \( 1 - 23.5T + 289T^{2} \)
19 \( 1 - 21.7iT - 361T^{2} \)
23 \( 1 - 30.7iT - 529T^{2} \)
29 \( 1 + 12.6T + 841T^{2} \)
31 \( 1 - 24.5iT - 961T^{2} \)
37 \( 1 + 18.2T + 1.36e3T^{2} \)
41 \( 1 - 38.0T + 1.68e3T^{2} \)
43 \( 1 + 34.9iT - 1.84e3T^{2} \)
47 \( 1 + 29.6iT - 2.20e3T^{2} \)
53 \( 1 + 39.3T + 2.80e3T^{2} \)
59 \( 1 + 65.3iT - 3.48e3T^{2} \)
61 \( 1 - 29.8T + 3.72e3T^{2} \)
67 \( 1 + 11.8iT - 4.48e3T^{2} \)
71 \( 1 - 140. iT - 5.04e3T^{2} \)
73 \( 1 - 119.T + 5.32e3T^{2} \)
79 \( 1 + 9.18iT - 6.24e3T^{2} \)
83 \( 1 + 113. iT - 6.88e3T^{2} \)
89 \( 1 - 7.88T + 7.92e3T^{2} \)
97 \( 1 - 55.5T + 9.40e3T^{2} \)
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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

−9.244468982966529138721258817979, −8.324263694564595874969976446239, −7.62601700407406526095699243053, −6.63323919466320222114316730105, −5.71645802005359564525045221828, −5.44670641245091962848691981441, −3.84852536504579357956639565898, −3.36722437340851111026292329038, −1.81870989066385292734661651505, −1.05835827692874315216956013028, 0.951974211211188949844246594516, 2.08386496587734030294022076838, 2.98253527947236713919839635130, 4.15702007443495061760026190023, 5.22643469173148103713410034252, 5.95401640101779527348931775087, 6.49189595374886431817795780481, 7.60980972445605752962990370092, 8.462454279312951190086764849723, 9.236562121597200113105609066017

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