Properties

Label 2-2112-1.1-c1-0-28
Degree $2$
Conductor $2112$
Sign $-1$
Analytic cond. $16.8644$
Root an. cond. $4.10662$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s + 9-s − 11-s − 4·13-s − 2·17-s − 2·21-s + 2·23-s − 5·25-s − 27-s + 2·29-s + 4·31-s + 33-s − 6·37-s + 4·39-s − 6·41-s − 12·43-s + 6·47-s − 3·49-s + 2·51-s + 2·63-s + 4·67-s − 2·69-s + 10·71-s + 2·73-s + 5·75-s − 2·77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.485·17-s − 0.436·21-s + 0.417·23-s − 25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.174·33-s − 0.986·37-s + 0.640·39-s − 0.937·41-s − 1.82·43-s + 0.875·47-s − 3/7·49-s + 0.280·51-s + 0.251·63-s + 0.488·67-s − 0.240·69-s + 1.18·71-s + 0.234·73-s + 0.577·75-s − 0.227·77-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $-1$
Analytic conductor: \(16.8644\)
Root analytic conductor: \(4.10662\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2112,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−8.588292958367558716534570992774, −7.942710865660897618647461812074, −7.09459379389830635630745773189, −6.42993764417006864713469932537, −5.24274058652077590095165528950, −4.93912381390666761861872331878, −3.90529526587309291887022370964, −2.61580326498876244659774387394, −1.58005593402553417518764847925, 0, 1.58005593402553417518764847925, 2.61580326498876244659774387394, 3.90529526587309291887022370964, 4.93912381390666761861872331878, 5.24274058652077590095165528950, 6.42993764417006864713469932537, 7.09459379389830635630745773189, 7.942710865660897618647461812074, 8.588292958367558716534570992774

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