Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 6·11-s − 2·13-s − 4·19-s − 6·23-s − 5·25-s + 6·29-s − 8·31-s − 2·37-s − 12·41-s − 4·43-s + 12·47-s + 49-s − 6·53-s + 10·61-s + 8·67-s + 6·71-s − 10·73-s − 6·77-s + 4·79-s + 12·83-s − 12·89-s + 2·91-s − 10·97-s − 12·101-s − 8·103-s + 6·107-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.80·11-s − 0.554·13-s − 0.917·19-s − 1.25·23-s − 25-s + 1.11·29-s − 1.43·31-s − 0.328·37-s − 1.87·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s + 1.28·61-s + 0.977·67-s + 0.712·71-s − 1.17·73-s − 0.683·77-s + 0.450·79-s + 1.31·83-s − 1.27·89-s + 0.209·91-s − 1.01·97-s − 1.19·101-s − 0.788·103-s + 0.580·107-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

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 4032,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.213446609484277420706681145366, −7.18052525439954131151018372494, −6.61214442273728283164556360378, −6.01676531608062216159532650200, −5.07643921282730531606229802841, −3.99784334743039531176458960720, −3.70585619782695729246427373887, −2.36003185284708036230417844213, −1.49990868331367069680950997074, 0, 1.49990868331367069680950997074, 2.36003185284708036230417844213, 3.70585619782695729246427373887, 3.99784334743039531176458960720, 5.07643921282730531606229802841, 6.01676531608062216159532650200, 6.61214442273728283164556360378, 7.18052525439954131151018372494, 8.213446609484277420706681145366

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