Properties

Label 2-4056-1.1-c1-0-37
Degree $2$
Conductor $4056$
Sign $-1$
Analytic cond. $32.3873$
Root an. cond. $5.69098$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 1.73·5-s − 2.73·7-s + 9-s − 2.73·11-s + 1.73·15-s − 0.267·17-s + 4.73·19-s + 2.73·21-s + 8.19·23-s − 2.00·25-s − 27-s + 7.92·29-s + 1.46·31-s + 2.73·33-s + 4.73·35-s − 1.53·37-s − 5·41-s + 12.1·43-s − 1.73·45-s + 3.26·47-s + 0.464·49-s + 0.267·51-s − 7.92·53-s + 4.73·55-s − 4.73·57-s − 6.26·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.774·5-s − 1.03·7-s + 0.333·9-s − 0.823·11-s + 0.447·15-s − 0.0649·17-s + 1.08·19-s + 0.596·21-s + 1.70·23-s − 0.400·25-s − 0.192·27-s + 1.47·29-s + 0.262·31-s + 0.475·33-s + 0.799·35-s − 0.252·37-s − 0.780·41-s + 1.85·43-s − 0.258·45-s + 0.476·47-s + 0.0663·49-s + 0.0375·51-s − 1.08·53-s + 0.638·55-s − 0.626·57-s − 0.802·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4056\)    =    \(2^{3} \cdot 3 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(32.3873\)
Root analytic conductor: \(5.69098\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4056,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T + 5T^{2} \)
7 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 + 2.73T + 11T^{2} \)
17 \( 1 + 0.267T + 17T^{2} \)
19 \( 1 - 4.73T + 19T^{2} \)
23 \( 1 - 8.19T + 23T^{2} \)
29 \( 1 - 7.92T + 29T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 + 1.53T + 37T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 - 3.26T + 47T^{2} \)
53 \( 1 + 7.92T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6.26T + 61T^{2} \)
67 \( 1 + 8.73T + 67T^{2} \)
71 \( 1 + 2.19T + 71T^{2} \)
73 \( 1 + 9.19T + 73T^{2} \)
79 \( 1 + 8.39T + 79T^{2} \)
83 \( 1 - 1.66T + 83T^{2} \)
89 \( 1 - 9.46T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−7.87514626430338417248699686501, −7.33834917040622634052807584106, −6.64166924215167527339639508484, −5.86844658257899849018064000441, −5.06001984708718948722267546863, −4.35744064203742990390267496171, −3.28244703044903084842834945063, −2.77467968877965928951046799271, −1.09206121676631554675627062125, 0, 1.09206121676631554675627062125, 2.77467968877965928951046799271, 3.28244703044903084842834945063, 4.35744064203742990390267496171, 5.06001984708718948722267546863, 5.86844658257899849018064000441, 6.64166924215167527339639508484, 7.33834917040622634052807584106, 7.87514626430338417248699686501

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