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

−8.141578011833829279170569074147, −7.40204984083045864235626196968, −6.74720425463568219864015542771, −5.65709425594782800081991625985, −5.06235319812411421082260238754, −4.51738819392819425389029556338, −3.12715321778064264925250360958, −2.66253667094540199353978149273, −1.42168355617734446021830350271, 0, 1.42168355617734446021830350271, 2.66253667094540199353978149273, 3.12715321778064264925250360958, 4.51738819392819425389029556338, 5.06235319812411421082260238754, 5.65709425594782800081991625985, 6.74720425463568219864015542771, 7.40204984083045864235626196968, 8.141578011833829279170569074147

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