Properties

Degree 2
Conductor $ 7 \cdot 11 \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·4-s − 5-s − 7-s + 9-s − 11-s − 4·12-s − 13-s − 2·15-s + 4·16-s + 2·17-s − 19-s + 2·20-s − 2·21-s − 5·23-s − 4·25-s − 4·27-s + 2·28-s − 5·29-s − 9·31-s − 2·33-s + 35-s − 2·36-s + 10·37-s − 2·39-s − 2·41-s − 13·43-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.15·12-s − 0.277·13-s − 0.516·15-s + 16-s + 0.485·17-s − 0.229·19-s + 0.447·20-s − 0.436·21-s − 1.04·23-s − 4/5·25-s − 0.769·27-s + 0.377·28-s − 0.928·29-s − 1.61·31-s − 0.348·33-s + 0.169·35-s − 1/3·36-s + 1.64·37-s − 0.320·39-s − 0.312·41-s − 1.98·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1001\)    =    \(7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1001} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1001,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−9.640083292373987058603191049812, −8.650195885178034734764412957721, −8.025597150713170782148421117427, −7.46595058593106128092987481128, −6.03687681304360526387090452910, −5.06188428719186619231283594875, −3.85872420740283031925332467940, −3.41704386442178853478657445265, −2.05343142591405820873295205760, 0, 2.05343142591405820873295205760, 3.41704386442178853478657445265, 3.85872420740283031925332467940, 5.06188428719186619231283594875, 6.03687681304360526387090452910, 7.46595058593106128092987481128, 8.025597150713170782148421117427, 8.650195885178034734764412957721, 9.640083292373987058603191049812

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