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 + 4·13-s − 6·17-s − 2·19-s − 5·25-s − 6·29-s − 4·31-s − 2·37-s − 6·41-s − 8·43-s + 12·47-s + 49-s + 6·53-s − 6·59-s − 8·61-s + 4·67-s + 2·73-s + 8·79-s − 6·83-s + 6·89-s + 4·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.10·13-s − 1.45·17-s − 0.458·19-s − 25-s − 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.781·59-s − 1.02·61-s + 0.488·67-s + 0.234·73-s + 0.900·79-s − 0.658·83-s + 0.635·89-s + 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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  :  $\chi_{4032} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−18.27711952428758, −17.96946632624893, −17.07023465155331, −16.75223536023259, −15.77259642172151, −15.38706293624591, −14.87295028917625, −13.87410022919720, −13.52412493160966, −12.93928230557644, −12.08373877127828, −11.39553218511717, −10.89739034779398, −10.32109817146828, −9.287321299743467, −8.820715669839055, −8.156759517304687, −7.348083910353564, −6.583386123703386, −5.893721156566383, −5.102768037472538, −4.143188175594794, −3.599914405449773, −2.311475981001490, −1.551949318390129, 0, 1.551949318390129, 2.311475981001490, 3.599914405449773, 4.143188175594794, 5.102768037472538, 5.893721156566383, 6.583386123703386, 7.348083910353564, 8.156759517304687, 8.820715669839055, 9.287321299743467, 10.32109817146828, 10.89739034779398, 11.39553218511717, 12.08373877127828, 12.93928230557644, 13.52412493160966, 13.87410022919720, 14.87295028917625, 15.38706293624591, 15.77259642172151, 16.75223536023259, 17.07023465155331, 17.96946632624893, 18.27711952428758

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