Properties

Label 2-12e3-1.1-c1-0-21
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 − 7-s + 2·11-s − 13-s + 6·17-s − 5·19-s + 6·23-s − 25-s − 8·29-s − 8·31-s + 2·35-s + 5·37-s + 8·41-s − 4·43-s − 10·47-s − 6·49-s − 4·53-s − 4·55-s − 14·59-s − 3·61-s + 2·65-s − 13·67-s − 4·71-s + 9·73-s − 2·77-s − 11·79-s + 12·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s + 0.603·11-s − 0.277·13-s + 1.45·17-s − 1.14·19-s + 1.25·23-s − 1/5·25-s − 1.48·29-s − 1.43·31-s + 0.338·35-s + 0.821·37-s + 1.24·41-s − 0.609·43-s − 1.45·47-s − 6/7·49-s − 0.549·53-s − 0.539·55-s − 1.82·59-s − 0.384·61-s + 0.248·65-s − 1.58·67-s − 0.474·71-s + 1.05·73-s − 0.227·77-s − 1.23·79-s + 1.31·83-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: \(1\)
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 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−9.091038044170517247847576338371, −7.908747723265166312221216125440, −7.53518840978743106177330231964, −6.56129874280569123918857312585, −5.72569558138507357075854697581, −4.68301032913957237602385420184, −3.77298647544947819109929811146, −3.09100156780521462109801579892, −1.58075024630036371346424385734, 0, 1.58075024630036371346424385734, 3.09100156780521462109801579892, 3.77298647544947819109929811146, 4.68301032913957237602385420184, 5.72569558138507357075854697581, 6.56129874280569123918857312585, 7.53518840978743106177330231964, 7.908747723265166312221216125440, 9.091038044170517247847576338371

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