Properties

Label 2-2112-1.1-c1-0-39
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s − 4·7-s + 9-s − 11-s − 6·13-s + 2·15-s + 6·17-s − 8·19-s − 4·21-s − 25-s + 27-s + 6·29-s − 33-s − 8·35-s − 6·37-s − 6·39-s − 10·41-s − 8·43-s + 2·45-s + 9·49-s + 6·51-s − 6·53-s − 2·55-s − 8·57-s + 4·59-s + 2·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.516·15-s + 1.45·17-s − 1.83·19-s − 0.872·21-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.174·33-s − 1.35·35-s − 0.986·37-s − 0.960·39-s − 1.56·41-s − 1.21·43-s + 0.298·45-s + 9/7·49-s + 0.840·51-s − 0.824·53-s − 0.269·55-s − 1.05·57-s + 0.520·59-s + 0.256·61-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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.791907514497419235916472892451, −8.017615825535372394970186433105, −6.99241400336604018651823749604, −6.49156479675218185201348443987, −5.57874719988269628577344168022, −4.70104962830372502873702609557, −3.47509279115318849018848825510, −2.76019896934793162407560091808, −1.88742699251523641900216036635, 0, 1.88742699251523641900216036635, 2.76019896934793162407560091808, 3.47509279115318849018848825510, 4.70104962830372502873702609557, 5.57874719988269628577344168022, 6.49156479675218185201348443987, 6.99241400336604018651823749604, 8.017615825535372394970186433105, 8.791907514497419235916472892451

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