Properties

Label 2-12e3-1.1-c1-0-13
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·7-s + 7·13-s + 19-s − 5·25-s − 4·31-s + 37-s − 8·43-s + 18·49-s + 13·61-s − 11·67-s + 17·73-s − 13·79-s + 35·91-s + 5·97-s − 7·103-s − 2·109-s + ⋯
L(s)  = 1  + 1.88·7-s + 1.94·13-s + 0.229·19-s − 25-s − 0.718·31-s + 0.164·37-s − 1.21·43-s + 18/7·49-s + 1.66·61-s − 1.34·67-s + 1.98·73-s − 1.46·79-s + 3.66·91-s + 0.507·97-s − 0.689·103-s − 0.191·109-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.360844297\)
\(L(\frac12)\) \(\approx\) \(2.360844297\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−9.125589838381877994041862400986, −8.360413074571016534853094652479, −8.001529503824122494641395863146, −7.02193571198558069600312241264, −5.94215330426161837619813679812, −5.28577670060288457890184502723, −4.31915400924746551561976811972, −3.53995560336243251785891738493, −2.01230440740899716446600775646, −1.20941721581339133701993176268, 1.20941721581339133701993176268, 2.01230440740899716446600775646, 3.53995560336243251785891738493, 4.31915400924746551561976811972, 5.28577670060288457890184502723, 5.94215330426161837619813679812, 7.02193571198558069600312241264, 8.001529503824122494641395863146, 8.360413074571016534853094652479, 9.125589838381877994041862400986

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