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Properties

Label 2-2112-88.43-c1-0-15
Degree $2$
Conductor $2112$
Sign $0.168 - 0.985i$
Analytic cond. $16.8644$
Root an. cond. $4.10662$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 1.05i·5-s + 1.97·7-s + 9-s + (−1.38 + 3.01i)11-s − 0.762·13-s + 1.05i·15-s + 4.17i·17-s + 0.233i·19-s + 1.97·21-s + 2.52i·23-s + 3.88·25-s + 27-s − 5.62·29-s − 4.88i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.472i·5-s + 0.745·7-s + 0.333·9-s + (−0.418 + 0.908i)11-s − 0.211·13-s + 0.272i·15-s + 1.01i·17-s + 0.0536i·19-s + 0.430·21-s + 0.525i·23-s + 0.776·25-s + 0.192·27-s − 1.04·29-s − 0.877i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $0.168 - 0.985i$
Analytic conductor: \(16.8644\)
Root analytic conductor: \(4.10662\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2112} (1759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2112,\ (\ :1/2),\ 0.168 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.106637888\)
\(L(\frac12)\) \(\approx\) \(2.106637888\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 - 3.01i)T \)
good5 \( 1 - 1.05iT - 5T^{2} \)
7 \( 1 - 1.97T + 7T^{2} \)
13 \( 1 + 0.762T + 13T^{2} \)
17 \( 1 - 4.17iT - 17T^{2} \)
19 \( 1 - 0.233iT - 19T^{2} \)
23 \( 1 - 2.52iT - 23T^{2} \)
29 \( 1 + 5.62T + 29T^{2} \)
31 \( 1 + 4.88iT - 31T^{2} \)
37 \( 1 - 2.69iT - 37T^{2} \)
41 \( 1 - 8.67iT - 41T^{2} \)
43 \( 1 - 8.12iT - 43T^{2} \)
47 \( 1 + 1.71iT - 47T^{2} \)
53 \( 1 + 2.60iT - 53T^{2} \)
59 \( 1 - 9.92T + 59T^{2} \)
61 \( 1 + 7.34T + 61T^{2} \)
67 \( 1 - 5.33T + 67T^{2} \)
71 \( 1 - 0.171iT - 71T^{2} \)
73 \( 1 + 2.88iT - 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 - 1.52iT - 83T^{2} \)
89 \( 1 - 8.92T + 89T^{2} \)
97 \( 1 + 0.565T + 97T^{2} \)
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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

−9.389458219221976414662682021390, −8.256698491236753905185405013668, −7.86612551057643498988987707839, −7.08322638530895278110730459393, −6.22751456262356330486767041594, −5.14485242826817016623296043312, −4.39714935899227359586102983266, −3.43220706335004486886470373346, −2.38888722212299006005864383781, −1.53822282815987665171445140373, 0.68984061654421094905123191109, 2.01142726700114263297828068257, 2.98840799850157872133902256163, 3.97659692244512612592675126978, 5.01495168331473759015306701351, 5.47212170793784007780763597253, 6.75626818980069702210373766146, 7.50131278088387648698849692084, 8.266801882849802052750147335204, 8.858224763669565301447866036649

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