Properties

Label 2-4032-1.1-c1-0-24
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 + 3.46·11-s − 2·13-s + 3.46·17-s − 4·19-s + 3.46·23-s + 6.99·25-s + 4·31-s − 3.46·35-s − 2·37-s + 10.3·41-s − 4·43-s − 6.92·47-s + 49-s + 6.92·53-s + 11.9·55-s − 6.92·59-s + 10·61-s − 6.92·65-s − 4·67-s + 10.3·71-s + 14·73-s − 3.46·77-s − 8·79-s + 11.9·85-s − 3.46·89-s + ⋯
L(s)  = 1  + 1.54·5-s − 0.377·7-s + 1.04·11-s − 0.554·13-s + 0.840·17-s − 0.917·19-s + 0.722·23-s + 1.39·25-s + 0.718·31-s − 0.585·35-s − 0.328·37-s + 1.62·41-s − 0.609·43-s − 1.01·47-s + 0.142·49-s + 0.951·53-s + 1.61·55-s − 0.901·59-s + 1.28·61-s − 0.859·65-s − 0.488·67-s + 1.23·71-s + 1.63·73-s − 0.394·77-s − 0.900·79-s + 1.30·85-s − 0.367·89-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: $\chi_{4032} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.740260628\)
\(L(\frac12)\) \(\approx\) \(2.740260628\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−8.676385520874038923855792276521, −7.65703016561002009349096065023, −6.67758690799875887313912039143, −6.34921579889703268347709557303, −5.54529508928040173858565299327, −4.82549226528423207535987336892, −3.80533589847814558665223621900, −2.79515025485135650029748623773, −1.98769202809105218884949704189, −0.997307287162411479574948561189, 0.997307287162411479574948561189, 1.98769202809105218884949704189, 2.79515025485135650029748623773, 3.80533589847814558665223621900, 4.82549226528423207535987336892, 5.54529508928040173858565299327, 6.34921579889703268347709557303, 6.67758690799875887313912039143, 7.65703016561002009349096065023, 8.676385520874038923855792276521

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