Properties

Label 2-12e3-1.1-c1-0-20
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  − 2·5-s − 3·7-s + 6·11-s + 3·13-s − 2·17-s − 3·19-s − 6·23-s − 25-s + 8·29-s + 6·35-s − 7·37-s + 8·41-s − 12·43-s − 6·47-s + 2·49-s − 4·53-s − 12·55-s + 6·59-s + 61-s − 6·65-s − 3·67-s − 12·71-s − 15·73-s − 18·77-s − 9·79-s − 12·83-s + 4·85-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.13·7-s + 1.80·11-s + 0.832·13-s − 0.485·17-s − 0.688·19-s − 1.25·23-s − 1/5·25-s + 1.48·29-s + 1.01·35-s − 1.15·37-s + 1.24·41-s − 1.82·43-s − 0.875·47-s + 2/7·49-s − 0.549·53-s − 1.61·55-s + 0.781·59-s + 0.128·61-s − 0.744·65-s − 0.366·67-s − 1.42·71-s − 1.75·73-s − 2.05·77-s − 1.01·79-s − 1.31·83-s + 0.433·85-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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−8.742871528992136639223499009719, −8.398804233680254223168104957402, −7.17330313041166053658325455178, −6.47142445887211239233670215635, −6.03993649285615255926110154202, −4.42534002049006891974002112914, −3.89073820849432411300555838274, −3.12135221332905635448639751702, −1.55247314198910365347987668608, 0, 1.55247314198910365347987668608, 3.12135221332905635448639751702, 3.89073820849432411300555838274, 4.42534002049006891974002112914, 6.03993649285615255926110154202, 6.47142445887211239233670215635, 7.17330313041166053658325455178, 8.398804233680254223168104957402, 8.742871528992136639223499009719

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