Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 2·4-s − 3·5-s + 6·6-s − 7-s + 6·9-s + 6·10-s + 11-s − 6·12-s − 13-s + 2·14-s + 9·15-s − 4·16-s − 8·17-s − 12·18-s − 4·19-s − 6·20-s + 3·21-s − 2·22-s − 9·23-s + 4·25-s + 2·26-s − 9·27-s − 2·28-s − 8·29-s − 18·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 4-s − 1.34·5-s + 2.44·6-s − 0.377·7-s + 2·9-s + 1.89·10-s + 0.301·11-s − 1.73·12-s − 0.277·13-s + 0.534·14-s + 2.32·15-s − 16-s − 1.94·17-s − 2.82·18-s − 0.917·19-s − 1.34·20-s + 0.654·21-s − 0.426·22-s − 1.87·23-s + 4/5·25-s + 0.392·26-s − 1.73·27-s − 0.377·28-s − 1.48·29-s − 3.28·30-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  =  2
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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−19.71873526936749, −19.43282891747278, −18.62675088714829, −17.95915336678862, −17.50546171678446, −16.77868116673423, −16.25597246785588, −15.69322262256624, −15.14111705715124, −13.57205251273787, −12.65544351695831, −11.86911218081885, −11.40625392413406, −10.75677556442916, −10.17315576690887, −9.177678656566866, −8.329663666284704, −7.423521439973927, −6.774886646510479, −6.036855255776760, −4.590409830086547, −4.059900154752037, −1.857212374791200, 0, 0, 1.857212374791200, 4.059900154752037, 4.590409830086547, 6.036855255776760, 6.774886646510479, 7.423521439973927, 8.329663666284704, 9.177678656566866, 10.17315576690887, 10.75677556442916, 11.40625392413406, 11.86911218081885, 12.65544351695831, 13.57205251273787, 15.14111705715124, 15.69322262256624, 16.25597246785588, 16.77868116673423, 17.50546171678446, 17.95915336678862, 18.62675088714829, 19.43282891747278, 19.71873526936749

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