Properties

Label 2-12e3-1.1-c1-0-11
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 3·7-s + 4·11-s − 13-s + 4·17-s + 19-s − 4·23-s + 11·25-s − 4·31-s − 12·35-s + 9·37-s + 8·43-s + 12·47-s + 2·49-s − 8·53-s + 16·55-s + 4·59-s + 5·61-s − 4·65-s − 11·67-s − 8·71-s + 73-s − 12·77-s − 5·79-s + 8·83-s + 16·85-s − 12·89-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.13·7-s + 1.20·11-s − 0.277·13-s + 0.970·17-s + 0.229·19-s − 0.834·23-s + 11/5·25-s − 0.718·31-s − 2.02·35-s + 1.47·37-s + 1.21·43-s + 1.75·47-s + 2/7·49-s − 1.09·53-s + 2.15·55-s + 0.520·59-s + 0.640·61-s − 0.496·65-s − 1.34·67-s − 0.949·71-s + 0.117·73-s − 1.36·77-s − 0.562·79-s + 0.878·83-s + 1.73·85-s − 1.27·89-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: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.338102627\)
\(L(\frac12)\) \(\approx\) \(2.338102627\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−9.398184286512178458263925867474, −8.961679325395712580723026277632, −7.59935786244490470822810350055, −6.71174548934044095208893948883, −5.98289624762003201649160243701, −5.65513065279316088246203234268, −4.30786919885967763645635100877, −3.22235141017577078764025505936, −2.26799229473408129186056872817, −1.13756958398551067927188738213, 1.13756958398551067927188738213, 2.26799229473408129186056872817, 3.22235141017577078764025505936, 4.30786919885967763645635100877, 5.65513065279316088246203234268, 5.98289624762003201649160243701, 6.71174548934044095208893948883, 7.59935786244490470822810350055, 8.961679325395712580723026277632, 9.398184286512178458263925867474

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