Properties

Label 2-2112-1.1-c1-0-2
Degree $2$
Conductor $2112$
Sign $1$
Analytic cond. $16.8644$
Root an. cond. $4.10662$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 0.508·5-s − 3.87·7-s + 9-s − 11-s − 0.508·13-s + 0.508·15-s + 7.36·17-s − 5.36·19-s + 3.87·21-s − 2.50·23-s − 4.74·25-s − 27-s − 7.36·29-s + 10.7·31-s + 33-s + 1.96·35-s − 8.72·37-s + 0.508·39-s + 8.37·41-s + 2.37·43-s − 0.508·45-s − 5.49·47-s + 7.98·49-s − 7.36·51-s + 12.2·53-s + 0.508·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.227·5-s − 1.46·7-s + 0.333·9-s − 0.301·11-s − 0.140·13-s + 0.131·15-s + 1.78·17-s − 1.23·19-s + 0.844·21-s − 0.522·23-s − 0.948·25-s − 0.192·27-s − 1.36·29-s + 1.92·31-s + 0.174·33-s + 0.332·35-s − 1.43·37-s + 0.0813·39-s + 1.30·41-s + 0.362·43-s − 0.0757·45-s − 0.801·47-s + 1.14·49-s − 1.03·51-s + 1.68·53-s + 0.0685·55-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2112,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8358051199\)
\(L(\frac12)\) \(\approx\) \(0.8358051199\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.508T + 5T^{2} \)
7 \( 1 + 3.87T + 7T^{2} \)
13 \( 1 + 0.508T + 13T^{2} \)
17 \( 1 - 7.36T + 17T^{2} \)
19 \( 1 + 5.36T + 19T^{2} \)
23 \( 1 + 2.50T + 23T^{2} \)
29 \( 1 + 7.36T + 29T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 + 8.72T + 37T^{2} \)
41 \( 1 - 8.37T + 41T^{2} \)
43 \( 1 - 2.37T + 43T^{2} \)
47 \( 1 + 5.49T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 - 1.01T + 59T^{2} \)
61 \( 1 - 7.23T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 - 1.14T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 3.01T + 97T^{2} \)
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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

−9.273163521100402610255170036629, −8.218707352185089056673963918054, −7.50414077249027763223520972066, −6.62105693040790114387840435168, −5.98186563073511129605729291692, −5.29876786227502208761274261281, −4.05581080728150155661605157838, −3.41934252425097818650714067311, −2.24806773282882522900664640154, −0.59575435269033480206262001868, 0.59575435269033480206262001868, 2.24806773282882522900664640154, 3.41934252425097818650714067311, 4.05581080728150155661605157838, 5.29876786227502208761274261281, 5.98186563073511129605729291692, 6.62105693040790114387840435168, 7.50414077249027763223520972066, 8.218707352185089056673963918054, 9.273163521100402610255170036629

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