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  :  $\chi_{1001} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
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

−19.79925990119436, −19.27149245071105, −18.43965812809061, −18.10072762767733, −16.93557336078159, −16.41882427788671, −15.35944402533974, −14.74276214637716, −14.28263020093669, −13.35795875369221, −13.03442979525305, −12.08497153491790, −11.15659212279503, −9.848645225474457, −9.640083292373987, −8.650195885178035, −8.025597150713171, −7.465950585931061, −6.036876813043605, −5.061884287191866, −3.858724207402830, −3.417043864421789, −2.053431425914058, 0, 2.053431425914058, 3.417043864421789, 3.858724207402830, 5.061884287191866, 6.036876813043605, 7.465950585931061, 8.025597150713171, 8.650195885178035, 9.640083292373987, 9.848645225474457, 11.15659212279503, 12.08497153491790, 13.03442979525305, 13.35795875369221, 14.28263020093669, 14.74276214637716, 15.35944402533974, 16.41882427788671, 16.93557336078159, 18.10072762767733, 18.43965812809061, 19.27149245071105, 19.79925990119436

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