Properties

Label 2-2112-1.1-c3-0-104
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 $1$

Origins

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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: \(1\)
Selberg data: \((2,\ 2112,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 719.T + 9.12e5T^{2} \)
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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

−8.472312525780441757180876742787, −7.42795288589861653139054629920, −6.65724078010004304756091399706, −5.91273849510928908582857892651, −5.21268579424062017996742294943, −4.64199298428006458455751959310, −3.22386322073883853001893178370, −2.10826661934345558133109046725, −1.43590741674930434303043348238, 0, 1.43590741674930434303043348238, 2.10826661934345558133109046725, 3.22386322073883853001893178370, 4.64199298428006458455751959310, 5.21268579424062017996742294943, 5.91273849510928908582857892651, 6.65724078010004304756091399706, 7.42795288589861653139054629920, 8.472312525780441757180876742787

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