Properties

Label 2-12e3-1.1-c1-0-19
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

−8.646897306767688973776858314960, −8.177655093885067516195802420137, −7.44444168021687832363479010767, −6.72818725632924944067815958116, −5.48544965844839209036398212504, −4.81849202265609017039962796294, −3.72379428045727412120464814901, −3.13977056925804084341234249361, −1.55100613792804333036729509228, 0, 1.55100613792804333036729509228, 3.13977056925804084341234249361, 3.72379428045727412120464814901, 4.81849202265609017039962796294, 5.48544965844839209036398212504, 6.72818725632924944067815958116, 7.44444168021687832363479010767, 8.177655093885067516195802420137, 8.646897306767688973776858314960

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