Properties

Label 2-12e3-1.1-c3-0-30
Degree $2$
Conductor $1728$
Sign $1$
Analytic cond. $101.955$
Root an. cond. $10.0972$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 15.4·5-s + 23.8·7-s + 14.2·11-s + 13.8·13-s + 80.5·17-s + 144.·19-s + 141.·23-s + 112.·25-s − 251.·29-s − 16.6·31-s − 367.·35-s − 305.·37-s − 429.·41-s + 181.·43-s + 79.4·47-s + 225.·49-s − 663.·53-s − 219.·55-s + 220.·59-s + 473.·61-s − 213.·65-s + 647.·67-s − 14.4·71-s + 776.·73-s + 339.·77-s − 257.·79-s + 1.28e3·83-s + ⋯
L(s)  = 1  − 1.37·5-s + 1.28·7-s + 0.390·11-s + 0.295·13-s + 1.14·17-s + 1.74·19-s + 1.27·23-s + 0.901·25-s − 1.60·29-s − 0.0965·31-s − 1.77·35-s − 1.35·37-s − 1.63·41-s + 0.644·43-s + 0.246·47-s + 0.655·49-s − 1.72·53-s − 0.538·55-s + 0.486·59-s + 0.993·61-s − 0.406·65-s + 1.18·67-s − 0.0242·71-s + 1.24·73-s + 0.502·77-s − 0.367·79-s + 1.69·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(101.955\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.208113255\)
\(L(\frac12)\) \(\approx\) \(2.208113255\)
\(L(\frac{5}{2})\) 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 \( 1 + 15.4T + 125T^{2} \)
7 \( 1 - 23.8T + 343T^{2} \)
11 \( 1 - 14.2T + 1.33e3T^{2} \)
13 \( 1 - 13.8T + 2.19e3T^{2} \)
17 \( 1 - 80.5T + 4.91e3T^{2} \)
19 \( 1 - 144.T + 6.85e3T^{2} \)
23 \( 1 - 141.T + 1.21e4T^{2} \)
29 \( 1 + 251.T + 2.43e4T^{2} \)
31 \( 1 + 16.6T + 2.97e4T^{2} \)
37 \( 1 + 305.T + 5.06e4T^{2} \)
41 \( 1 + 429.T + 6.89e4T^{2} \)
43 \( 1 - 181.T + 7.95e4T^{2} \)
47 \( 1 - 79.4T + 1.03e5T^{2} \)
53 \( 1 + 663.T + 1.48e5T^{2} \)
59 \( 1 - 220.T + 2.05e5T^{2} \)
61 \( 1 - 473.T + 2.26e5T^{2} \)
67 \( 1 - 647.T + 3.00e5T^{2} \)
71 \( 1 + 14.4T + 3.57e5T^{2} \)
73 \( 1 - 776.T + 3.89e5T^{2} \)
79 \( 1 + 257.T + 4.93e5T^{2} \)
83 \( 1 - 1.28e3T + 5.71e5T^{2} \)
89 \( 1 - 156.T + 7.04e5T^{2} \)
97 \( 1 - 1.16e3T + 9.12e5T^{2} \)
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

−8.824175350331989322194491701653, −7.962329316961856912054696509992, −7.59964695747235742277194189476, −6.84696376979825856880117968712, −5.35632669414955620631450574763, −5.00070823066020826338269587984, −3.75593557045953240798197042368, −3.31639389024373499978039220770, −1.65376343970714549612173030762, −0.75841012413738186725470397206, 0.75841012413738186725470397206, 1.65376343970714549612173030762, 3.31639389024373499978039220770, 3.75593557045953240798197042368, 5.00070823066020826338269587984, 5.35632669414955620631450574763, 6.84696376979825856880117968712, 7.59964695747235742277194189476, 7.962329316961856912054696509992, 8.824175350331989322194491701653

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