Properties

Label 2-4032-1.1-c1-0-37
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  − 4·5-s + 7-s + 2·17-s + 2·19-s − 8·23-s + 11·25-s + 2·29-s + 4·31-s − 4·35-s + 6·37-s + 2·41-s − 8·43-s + 4·47-s + 49-s − 10·53-s + 6·59-s − 4·61-s + 12·67-s − 14·73-s − 8·79-s + 6·83-s − 8·85-s − 10·89-s − 8·95-s − 2·97-s + 12·101-s − 12·103-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.377·7-s + 0.485·17-s + 0.458·19-s − 1.66·23-s + 11/5·25-s + 0.371·29-s + 0.718·31-s − 0.676·35-s + 0.986·37-s + 0.312·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 1.37·53-s + 0.781·59-s − 0.512·61-s + 1.46·67-s − 1.63·73-s − 0.900·79-s + 0.658·83-s − 0.867·85-s − 1.05·89-s − 0.820·95-s − 0.203·97-s + 1.19·101-s − 1.18·103-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: $\chi_{4032} (1, \cdot )$
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.092660011067177558429540865439, −7.54801754874801697338809881828, −6.77420851319528696059978803031, −5.85675979257035438626438847546, −4.85762247801381754805120738895, −4.20793108421579356205513357787, −3.56270542712905614063510381286, −2.65043598386331574261041648988, −1.21717442621058177691884331097, 0, 1.21717442621058177691884331097, 2.65043598386331574261041648988, 3.56270542712905614063510381286, 4.20793108421579356205513357787, 4.85762247801381754805120738895, 5.85675979257035438626438847546, 6.77420851319528696059978803031, 7.54801754874801697338809881828, 8.092660011067177558429540865439

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