Properties

Label 2-4032-1.1-c1-0-56
Degree $2$
Conductor $4032$
Sign $-1$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 7-s − 2·11-s − 2·13-s − 6·17-s − 4·19-s + 6·23-s − 25-s + 4·31-s + 2·35-s − 10·37-s − 2·41-s − 4·43-s + 4·47-s + 49-s − 12·53-s − 4·55-s − 12·59-s − 6·61-s − 4·65-s − 4·67-s − 14·71-s − 2·73-s − 2·77-s + 8·79-s + 16·83-s − 12·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 0.603·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1.25·23-s − 1/5·25-s + 0.718·31-s + 0.338·35-s − 1.64·37-s − 0.312·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 1.64·53-s − 0.539·55-s − 1.56·59-s − 0.768·61-s − 0.496·65-s − 0.488·67-s − 1.66·71-s − 0.234·73-s − 0.227·77-s + 0.900·79-s + 1.75·83-s − 1.30·85-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.141634081829378081441809236752, −7.28267859617118153010735912842, −6.56064108292960750609847244365, −5.92082892086786754586742253811, −4.89364263062340315050988534287, −4.61650350267300135260926956930, −3.24669297328856350146221556739, −2.33542067219372894622175613585, −1.65405632235840502550099569734, 0, 1.65405632235840502550099569734, 2.33542067219372894622175613585, 3.24669297328856350146221556739, 4.61650350267300135260926956930, 4.89364263062340315050988534287, 5.92082892086786754586742253811, 6.56064108292960750609847244365, 7.28267859617118153010735912842, 8.141634081829378081441809236752

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