Properties

Label 2-12e3-1.1-c1-0-27
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

Related objects

Downloads

Learn more

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 \( 1 - 9 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.003152340707084154197169692141, −8.224131195573790842679615339585, −7.24143476508618264591640621751, −6.45918568156737461680179191076, −5.63116604718208874172193979497, −5.10278218324330686292022431163, −3.62018299831604404971780816203, −2.87665746571365337045650868782, −1.78875929979941638272154943927, 0, 1.78875929979941638272154943927, 2.87665746571365337045650868782, 3.62018299831604404971780816203, 5.10278218324330686292022431163, 5.63116604718208874172193979497, 6.45918568156737461680179191076, 7.24143476508618264591640621751, 8.224131195573790842679615339585, 9.003152340707084154197169692141

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