Properties

Label 2-12e3-12.11-c1-0-0
Degree $2$
Conductor $1728$
Sign $-1$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.86i·5-s − 3.73i·7-s + 1.03·11-s − 4.46·13-s + 1.79i·17-s − 1.73i·19-s − 8.76·23-s − 9.92·25-s + 7.72i·29-s + 7.46i·31-s + 14.4·35-s − 0.464·37-s − 7.72i·41-s − 0.535i·43-s + 4.62·47-s + ⋯
L(s)  = 1  + 1.72i·5-s − 1.41i·7-s + 0.312·11-s − 1.23·13-s + 0.434i·17-s − 0.397i·19-s − 1.82·23-s − 1.98·25-s + 1.43i·29-s + 1.34i·31-s + 2.43·35-s − 0.0762·37-s − 1.20i·41-s − 0.0817i·43-s + 0.674·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3717954275\)
\(L(\frac12)\) \(\approx\) \(0.3717954275\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.86iT - 5T^{2} \)
7 \( 1 + 3.73iT - 7T^{2} \)
11 \( 1 - 1.03T + 11T^{2} \)
13 \( 1 + 4.46T + 13T^{2} \)
17 \( 1 - 1.79iT - 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + 8.76T + 23T^{2} \)
29 \( 1 - 7.72iT - 29T^{2} \)
31 \( 1 - 7.46iT - 31T^{2} \)
37 \( 1 + 0.464T + 37T^{2} \)
41 \( 1 + 7.72iT - 41T^{2} \)
43 \( 1 + 0.535iT - 43T^{2} \)
47 \( 1 - 4.62T + 47T^{2} \)
53 \( 1 + 3.58iT - 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 6.26iT - 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 3.92T + 73T^{2} \)
79 \( 1 + 4.80iT - 79T^{2} \)
83 \( 1 - 2.07T + 83T^{2} \)
89 \( 1 - 1.79iT - 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−10.12110240642347628165865829331, −9.016830191581566202281026416719, −7.77442974069517781535884713316, −7.22777681258713777197601575574, −6.79429916450995450170649735382, −5.89286934689672001095346581120, −4.60362482453938919280932612821, −3.73220579314558851810244513061, −2.97748860867132492979997480728, −1.79994655036323348597449368478, 0.12956246680418703808014698963, 1.73006087471869684524593458834, 2.59449880349643795372450009826, 4.19242323428476113845165167264, 4.76452222679945403365421342063, 5.72340102619177249711081065459, 6.11969137705100033316852160407, 7.81348796564509402950401054947, 8.025896335713027600125690980472, 9.220207457546641892834296354527

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