Properties

Label 2-2112-1.1-c3-0-110
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 + 7.12·5-s + 9.12·7-s + 9·9-s − 11·11-s − 69.0·13-s + 21.3·15-s − 126.·17-s + 121.·19-s + 27.3·21-s + 2.60·23-s − 74.2·25-s + 27·27-s + 88.7·29-s + 132.·31-s − 33·33-s + 64.9·35-s + 411.·37-s − 207.·39-s − 283.·41-s − 364.·43-s + 64.1·45-s − 342.·47-s − 259.·49-s − 378.·51-s + 343.·53-s − 78.3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.637·5-s + 0.492·7-s + 0.333·9-s − 0.301·11-s − 1.47·13-s + 0.367·15-s − 1.80·17-s + 1.46·19-s + 0.284·21-s + 0.0235·23-s − 0.594·25-s + 0.192·27-s + 0.568·29-s + 0.770·31-s − 0.174·33-s + 0.313·35-s + 1.82·37-s − 0.851·39-s − 1.08·41-s − 1.29·43-s + 0.212·45-s − 1.06·47-s − 0.757·49-s − 1.03·51-s + 0.890·53-s − 0.192·55-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 \( 1 - 7.12T + 125T^{2} \)
7 \( 1 - 9.12T + 343T^{2} \)
13 \( 1 + 69.0T + 2.19e3T^{2} \)
17 \( 1 + 126.T + 4.91e3T^{2} \)
19 \( 1 - 121.T + 6.85e3T^{2} \)
23 \( 1 - 2.60T + 1.21e4T^{2} \)
29 \( 1 - 88.7T + 2.43e4T^{2} \)
31 \( 1 - 132.T + 2.97e4T^{2} \)
37 \( 1 - 411.T + 5.06e4T^{2} \)
41 \( 1 + 283.T + 6.89e4T^{2} \)
43 \( 1 + 364.T + 7.95e4T^{2} \)
47 \( 1 + 342.T + 1.03e5T^{2} \)
53 \( 1 - 343.T + 1.48e5T^{2} \)
59 \( 1 + 493.T + 2.05e5T^{2} \)
61 \( 1 + 619.T + 2.26e5T^{2} \)
67 \( 1 - 452.T + 3.00e5T^{2} \)
71 \( 1 + 698.T + 3.57e5T^{2} \)
73 \( 1 + 793.T + 3.89e5T^{2} \)
79 \( 1 + 340.T + 4.93e5T^{2} \)
83 \( 1 + 1.12e3T + 5.71e5T^{2} \)
89 \( 1 + 185.T + 7.04e5T^{2} \)
97 \( 1 - 1.14e3T + 9.12e5T^{2} \)
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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

−8.337489322325471838184407047708, −7.63677027791231459218617039427, −6.90668145349513439294869556249, −6.01802430917160566364053382828, −4.89900040533400795602638769789, −4.54218446818073116428382117056, −3.08752555256402779561180415397, −2.38038214960761178044268491074, −1.50810991038328436082166500723, 0, 1.50810991038328436082166500723, 2.38038214960761178044268491074, 3.08752555256402779561180415397, 4.54218446818073116428382117056, 4.89900040533400795602638769789, 6.01802430917160566364053382828, 6.90668145349513439294869556249, 7.63677027791231459218617039427, 8.337489322325471838184407047708

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