Properties

Label 2-2112-1.1-c3-0-52
Degree $2$
Conductor $2112$
Sign $1$
Analytic cond. $124.612$
Root an. cond. $11.1629$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 16.8·5-s − 7.69·7-s + 9·9-s + 11·11-s − 24.8·13-s + 50.5·15-s − 15.9·17-s − 15.1·19-s − 23.0·21-s + 17.7·23-s + 158.·25-s + 27·27-s + 128.·29-s + 219.·31-s + 33·33-s − 129.·35-s − 92.0·37-s − 74.5·39-s − 459.·41-s − 64.9·43-s + 151.·45-s + 497.·47-s − 283.·49-s − 47.8·51-s + 526.·53-s + 185.·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.50·5-s − 0.415·7-s + 0.333·9-s + 0.301·11-s − 0.530·13-s + 0.870·15-s − 0.227·17-s − 0.182·19-s − 0.239·21-s + 0.160·23-s + 1.27·25-s + 0.192·27-s + 0.823·29-s + 1.27·31-s + 0.174·33-s − 0.626·35-s − 0.409·37-s − 0.306·39-s − 1.75·41-s − 0.230·43-s + 0.502·45-s + 1.54·47-s − 0.827·49-s − 0.131·51-s + 1.36·53-s + 0.454·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $1$
Analytic conductor: \(124.612\)
Root analytic conductor: \(11.1629\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2112,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.856760070\)
\(L(\frac12)\) \(\approx\) \(3.856760070\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
11 \( 1 - 11T \)
good5 \( 1 - 16.8T + 125T^{2} \)
7 \( 1 + 7.69T + 343T^{2} \)
13 \( 1 + 24.8T + 2.19e3T^{2} \)
17 \( 1 + 15.9T + 4.91e3T^{2} \)
19 \( 1 + 15.1T + 6.85e3T^{2} \)
23 \( 1 - 17.7T + 1.21e4T^{2} \)
29 \( 1 - 128.T + 2.43e4T^{2} \)
31 \( 1 - 219.T + 2.97e4T^{2} \)
37 \( 1 + 92.0T + 5.06e4T^{2} \)
41 \( 1 + 459.T + 6.89e4T^{2} \)
43 \( 1 + 64.9T + 7.95e4T^{2} \)
47 \( 1 - 497.T + 1.03e5T^{2} \)
53 \( 1 - 526.T + 1.48e5T^{2} \)
59 \( 1 - 578.T + 2.05e5T^{2} \)
61 \( 1 - 221.T + 2.26e5T^{2} \)
67 \( 1 - 860.T + 3.00e5T^{2} \)
71 \( 1 - 580.T + 3.57e5T^{2} \)
73 \( 1 - 510.T + 3.89e5T^{2} \)
79 \( 1 - 1.03e3T + 4.93e5T^{2} \)
83 \( 1 + 606.T + 5.71e5T^{2} \)
89 \( 1 + 23.4T + 7.04e5T^{2} \)
97 \( 1 - 719.T + 9.12e5T^{2} \)
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

−8.787955554506152727636082600035, −8.201963627379616903073662656232, −6.87952869355335966258869322050, −6.59675867310218615932553151803, −5.56423545475154584205547581036, −4.83079578352825448372598961068, −3.69297012146595018938153208593, −2.63627321239657181102106115026, −2.03817420668938760587264969674, −0.880693359903389549638154132700, 0.880693359903389549638154132700, 2.03817420668938760587264969674, 2.63627321239657181102106115026, 3.69297012146595018938153208593, 4.83079578352825448372598961068, 5.56423545475154584205547581036, 6.59675867310218615932553151803, 6.87952869355335966258869322050, 8.201963627379616903073662656232, 8.787955554506152727636082600035

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