Properties

Label 2-2112-1.1-c3-0-60
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 2.84·5-s + 31.6·7-s + 9·9-s + 11·11-s − 5.15·13-s − 8.54·15-s + 121.·17-s − 34.8·19-s + 95.0·21-s + 116.·23-s − 116.·25-s + 27·27-s + 69.4·29-s + 140.·31-s + 33·33-s − 90.3·35-s + 420.·37-s − 15.4·39-s − 322.·41-s − 321.·43-s − 25.6·45-s − 231.·47-s + 661.·49-s + 365.·51-s − 4.91·53-s − 31.3·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.254·5-s + 1.71·7-s + 0.333·9-s + 0.301·11-s − 0.109·13-s − 0.147·15-s + 1.73·17-s − 0.420·19-s + 0.988·21-s + 1.05·23-s − 0.935·25-s + 0.192·27-s + 0.444·29-s + 0.814·31-s + 0.174·33-s − 0.436·35-s + 1.86·37-s − 0.0634·39-s − 1.22·41-s − 1.13·43-s − 0.0849·45-s − 0.718·47-s + 1.92·49-s + 1.00·51-s − 0.0127·53-s − 0.0768·55-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

\(L(2)\) \(\approx\) \(3.969081516\)
\(L(\frac12)\) \(\approx\) \(3.969081516\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.84T + 125T^{2} \)
7 \( 1 - 31.6T + 343T^{2} \)
13 \( 1 + 5.15T + 2.19e3T^{2} \)
17 \( 1 - 121.T + 4.91e3T^{2} \)
19 \( 1 + 34.8T + 6.85e3T^{2} \)
23 \( 1 - 116.T + 1.21e4T^{2} \)
29 \( 1 - 69.4T + 2.43e4T^{2} \)
31 \( 1 - 140.T + 2.97e4T^{2} \)
37 \( 1 - 420.T + 5.06e4T^{2} \)
41 \( 1 + 322.T + 6.89e4T^{2} \)
43 \( 1 + 321.T + 7.95e4T^{2} \)
47 \( 1 + 231.T + 1.03e5T^{2} \)
53 \( 1 + 4.91T + 1.48e5T^{2} \)
59 \( 1 + 406.T + 2.05e5T^{2} \)
61 \( 1 - 556.T + 2.26e5T^{2} \)
67 \( 1 + 84.7T + 3.00e5T^{2} \)
71 \( 1 - 49.0T + 3.57e5T^{2} \)
73 \( 1 - 785.T + 3.89e5T^{2} \)
79 \( 1 + 383.T + 4.93e5T^{2} \)
83 \( 1 - 930.T + 5.71e5T^{2} \)
89 \( 1 + 732.T + 7.04e5T^{2} \)
97 \( 1 + 1.17e3T + 9.12e5T^{2} \)
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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

−8.428346976062768568731586602970, −8.082097318491430734046675887409, −7.49450151294612808525368414133, −6.48304156547927639540138723287, −5.34743954098622529655955247845, −4.72269288796831815158821058234, −3.84597992247146639688262332561, −2.85513510235764174783428250135, −1.73398884360358607943324961714, −0.965698286353920623451203490342, 0.965698286353920623451203490342, 1.73398884360358607943324961714, 2.85513510235764174783428250135, 3.84597992247146639688262332561, 4.72269288796831815158821058234, 5.34743954098622529655955247845, 6.48304156547927639540138723287, 7.49450151294612808525368414133, 8.082097318491430734046675887409, 8.428346976062768568731586602970

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