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Properties

Label 2-12e3-3.2-c0-0-0
Degree $2$
Conductor $1728$
Sign $1$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 13-s + 19-s + 25-s + 2·31-s + 37-s − 2·43-s + 61-s + 67-s − 73-s − 79-s − 91-s − 97-s − 103-s − 2·109-s + ⋯
L(s)  = 1  − 7-s + 13-s + 19-s + 25-s + 2·31-s + 37-s − 2·43-s + 61-s + 67-s − 73-s − 79-s − 91-s − 97-s − 103-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1728} (1025, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.092671086\)
\(L(\frac12)\) \(\approx\) \(1.092671086\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 - T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 - T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )^{2} \)
37 \( 1 - T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−9.676176891036334364272920520262, −8.670453305722239995493802645981, −8.081417999545092363500546419982, −6.92058377856422659273126683379, −6.42808152131346192290157862576, −5.54280158641685867684558932946, −4.52736452858278073357114034117, −3.45834322318940291603940970882, −2.77535106940288001937305470671, −1.14418608506021133181936611859, 1.14418608506021133181936611859, 2.77535106940288001937305470671, 3.45834322318940291603940970882, 4.52736452858278073357114034117, 5.54280158641685867684558932946, 6.42808152131346192290157862576, 6.92058377856422659273126683379, 8.081417999545092363500546419982, 8.670453305722239995493802645981, 9.676176891036334364272920520262

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