Properties

Label 2-8112-1.1-c1-0-86
Degree $2$
Conductor $8112$
Sign $1$
Analytic cond. $64.7746$
Root an. cond. $8.04826$
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 + 3.46·7-s + 9-s + 3.46·11-s + 6·17-s + 3.46·19-s + 3.46·21-s − 5·25-s + 27-s + 6·29-s − 3.46·31-s + 3.46·33-s + 6.92·37-s + 6.92·41-s − 4·43-s − 3.46·47-s + 4.99·49-s + 6·51-s + 6·53-s + 3.46·57-s − 10.3·59-s − 2·61-s + 3.46·63-s − 10.3·67-s + 3.46·71-s − 5·75-s + 11.9·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.30·7-s + 0.333·9-s + 1.04·11-s + 1.45·17-s + 0.794·19-s + 0.755·21-s − 25-s + 0.192·27-s + 1.11·29-s − 0.622·31-s + 0.603·33-s + 1.13·37-s + 1.08·41-s − 0.609·43-s − 0.505·47-s + 0.714·49-s + 0.840·51-s + 0.824·53-s + 0.458·57-s − 1.35·59-s − 0.256·61-s + 0.436·63-s − 1.26·67-s + 0.411·71-s − 0.577·75-s + 1.36·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8112\)    =    \(2^{4} \cdot 3 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(64.7746\)
Root analytic conductor: \(8.04826\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8112,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.852697475\)
\(L(\frac12)\) \(\approx\) \(3.852697475\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + 6.92T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−7.75979783165705424566202795667, −7.47936210668792377399126137959, −6.44021454069055615981741747992, −5.71320085375837457308536366282, −4.95544666487255531877705201633, −4.23447944588114909919764725444, −3.54401290735212644198883177137, −2.68087828813692868206999461218, −1.61630410865584575575365154290, −1.08086567169714678337704020569, 1.08086567169714678337704020569, 1.61630410865584575575365154290, 2.68087828813692868206999461218, 3.54401290735212644198883177137, 4.23447944588114909919764725444, 4.95544666487255531877705201633, 5.71320085375837457308536366282, 6.44021454069055615981741747992, 7.47936210668792377399126137959, 7.75979783165705424566202795667

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