Properties

Label 2-4032-1.1-c1-0-27
Degree $2$
Conductor $4032$
Sign $1$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
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  + 3.46·5-s + 7-s + 1.46·11-s − 2·13-s − 0.535·17-s + 6.92·19-s − 1.46·23-s + 6.99·25-s − 4.92·29-s + 10.9·31-s + 3.46·35-s + 2·37-s − 11.4·41-s − 8·43-s + 10.9·47-s + 49-s − 2·53-s + 5.07·55-s + 1.07·59-s + 8.92·61-s − 6.92·65-s − 2.92·67-s + 9.46·71-s + 12.9·73-s + 1.46·77-s + 10.9·79-s + 4·83-s + ⋯
L(s)  = 1  + 1.54·5-s + 0.377·7-s + 0.441·11-s − 0.554·13-s − 0.129·17-s + 1.58·19-s − 0.305·23-s + 1.39·25-s − 0.915·29-s + 1.96·31-s + 0.585·35-s + 0.328·37-s − 1.79·41-s − 1.21·43-s + 1.59·47-s + 0.142·49-s − 0.274·53-s + 0.683·55-s + 0.139·59-s + 1.14·61-s − 0.859·65-s − 0.357·67-s + 1.12·71-s + 1.51·73-s + 0.166·77-s + 1.22·79-s + 0.439·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.951361529\)
\(L(\frac12)\) \(\approx\) \(2.951361529\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 0.535T + 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 + 1.46T + 23T^{2} \)
29 \( 1 + 4.92T + 29T^{2} \)
31 \( 1 - 10.9T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 1.07T + 59T^{2} \)
61 \( 1 - 8.92T + 61T^{2} \)
67 \( 1 + 2.92T + 67T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 + 8.92T + 97T^{2} \)
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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

−8.509672252713730275777334532126, −7.71787817246087997202866334029, −6.81623333407414810570331582483, −6.27419421997996820185455579567, −5.30770811522963424351480483843, −5.04030801807626924344360398248, −3.80925427426975247268674329409, −2.75783847575321147976885301170, −1.96121602390600694359033349434, −1.05501993262064266688989517679, 1.05501993262064266688989517679, 1.96121602390600694359033349434, 2.75783847575321147976885301170, 3.80925427426975247268674329409, 5.04030801807626924344360398248, 5.30770811522963424351480483843, 6.27419421997996820185455579567, 6.81623333407414810570331582483, 7.71787817246087997202866334029, 8.509672252713730275777334532126

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