Properties

Degree 2
Conductor $ 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  − 3-s − 2·4-s − 5-s − 2·7-s − 2·9-s − 11-s + 2·12-s − 13-s + 15-s + 4·16-s − 4·17-s + 2·19-s + 2·20-s + 2·21-s + 7·23-s − 4·25-s + 5·27-s + 4·28-s − 2·29-s − 3·31-s + 33-s + 2·35-s + 4·36-s − 11·37-s + 39-s + 10·41-s − 4·43-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s − 0.755·7-s − 2/3·9-s − 0.301·11-s + 0.577·12-s − 0.277·13-s + 0.258·15-s + 16-s − 0.970·17-s + 0.458·19-s + 0.447·20-s + 0.436·21-s + 1.45·23-s − 4/5·25-s + 0.962·27-s + 0.755·28-s − 0.371·29-s − 0.538·31-s + 0.174·33-s + 0.338·35-s + 2/3·36-s − 1.80·37-s + 0.160·39-s + 1.56·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(143\)    =    \(11 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{143} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 143,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{11,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−19.64980866399318, −19.17462857833665, −18.03933830335883, −17.40354613049151, −16.52913167322781, −15.55109773399813, −14.50303417159336, −13.43644081795268, −12.69739775523993, −11.63719964346043, −10.62154482736632, −9.421092482696968, −8.585957272776804, −7.216612243387683, −5.821358493393874, −4.757680415334523, −3.293045946759516, 0, 3.293045946759516, 4.757680415334523, 5.821358493393874, 7.216612243387683, 8.585957272776804, 9.421092482696968, 10.62154482736632, 11.63719964346043, 12.69739775523993, 13.43644081795268, 14.50303417159336, 15.55109773399813, 16.52913167322781, 17.40354613049151, 18.03933830335883, 19.17462857833665, 19.64980866399318

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