Properties

Degree 2
Conductor 89
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-s − 3-s − 4-s − 5-s + 6-s − 4·7-s + 3·8-s − 2·9-s + 10-s − 2·11-s + 12-s + 2·13-s + 4·14-s + 15-s − 16-s + 3·17-s + 2·18-s − 5·19-s + 20-s + 4·21-s + 2·22-s + 7·23-s − 3·24-s − 4·25-s − 2·26-s + 5·27-s + 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s + 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.14·19-s + 0.223·20-s + 0.872·21-s + 0.426·22-s + 1.45·23-s − 0.612·24-s − 4/5·25-s − 0.392·26-s + 0.962·27-s + 0.755·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{89} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 89,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 89$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 89$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad89 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−19.21887781517451, −18.68633383350932, −17.55989995998317, −16.66589070633432, −16.11254316403334, −14.76380738041646, −13.31042167272302, −12.68407335930173, −11.21974022600405, −10.27280155411413, −9.201725180013692, −8.158202871876967, −6.708284981985063, −5.336557794218001, −3.509791103812151, 0, 3.509791103812151, 5.336557794218001, 6.708284981985063, 8.158202871876967, 9.201725180013692, 10.27280155411413, 11.21974022600405, 12.68407335930173, 13.31042167272302, 14.76380738041646, 16.11254316403334, 16.66589070633432, 17.55989995998317, 18.68633383350932, 19.21887781517451

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