Properties

Label 2-12e3-1.1-c1-0-18
Degree $2$
Conductor $1728$
Sign $-1$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 7-s + 3·11-s + 4·13-s − 2·19-s − 6·23-s + 4·25-s − 6·29-s + 5·31-s + 3·35-s − 2·37-s − 6·41-s + 10·43-s + 6·47-s − 6·49-s − 9·53-s − 9·55-s − 12·59-s − 8·61-s − 12·65-s − 14·67-s − 7·73-s − 3·77-s + 8·79-s + 3·83-s − 18·89-s − 4·91-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s + 0.904·11-s + 1.10·13-s − 0.458·19-s − 1.25·23-s + 4/5·25-s − 1.11·29-s + 0.898·31-s + 0.507·35-s − 0.328·37-s − 0.937·41-s + 1.52·43-s + 0.875·47-s − 6/7·49-s − 1.23·53-s − 1.21·55-s − 1.56·59-s − 1.02·61-s − 1.48·65-s − 1.71·67-s − 0.819·73-s − 0.341·77-s + 0.900·79-s + 0.329·83-s − 1.90·89-s − 0.419·91-s + ⋯

Functional equation

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

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: yes
Arithmetic: yes
Character: $\chi_{1728} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−8.853339056894194014756168928413, −8.104088285674841823256361524017, −7.46895104625204893367477999066, −6.47796475592373752838072674203, −5.89467970712550736408970851242, −4.43365380485938957073214377677, −3.92407328889816322387260499130, −3.12927788056823217499419163319, −1.52887349549650490954756981970, 0, 1.52887349549650490954756981970, 3.12927788056823217499419163319, 3.92407328889816322387260499130, 4.43365380485938957073214377677, 5.89467970712550736408970851242, 6.47796475592373752838072674203, 7.46895104625204893367477999066, 8.104088285674841823256361524017, 8.853339056894194014756168928413

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