Properties

Label 2-12e3-1.1-c1-0-2
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  − 5-s − 3·7-s − 5·11-s − 4·13-s + 8·17-s + 2·19-s + 2·23-s − 4·25-s + 6·29-s + 7·31-s + 3·35-s + 6·37-s + 6·41-s − 2·43-s + 6·47-s + 2·49-s + 5·53-s + 5·55-s + 4·59-s + 8·61-s + 4·65-s − 10·67-s − 8·71-s + 73-s + 15·77-s − 16·79-s + 11·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s − 1.50·11-s − 1.10·13-s + 1.94·17-s + 0.458·19-s + 0.417·23-s − 4/5·25-s + 1.11·29-s + 1.25·31-s + 0.507·35-s + 0.986·37-s + 0.937·41-s − 0.304·43-s + 0.875·47-s + 2/7·49-s + 0.686·53-s + 0.674·55-s + 0.520·59-s + 1.02·61-s + 0.496·65-s − 1.22·67-s − 0.949·71-s + 0.117·73-s + 1.70·77-s − 1.80·79-s + 1.20·83-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

\(L(1)\) \(\approx\) \(1.069689778\)
\(L(\frac12)\) \(\approx\) \(1.069689778\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−9.649745872944478294591396163713, −8.352104784585146468182630520222, −7.67996846591970940803333630612, −7.15340015105356378963058851080, −5.98276615954030285758204092167, −5.32932694357691635729819844111, −4.34190910531857724149474492775, −3.11556422792398663255142778514, −2.66387606752889082799803334542, −0.68688623656136131006877383651, 0.68688623656136131006877383651, 2.66387606752889082799803334542, 3.11556422792398663255142778514, 4.34190910531857724149474492775, 5.32932694357691635729819844111, 5.98276615954030285758204092167, 7.15340015105356378963058851080, 7.67996846591970940803333630612, 8.352104784585146468182630520222, 9.649745872944478294591396163713

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