Properties

Degree $2$
Conductor $89$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s − 4-s − 2·5-s + 2·6-s + 2·7-s − 3·8-s + 9-s − 2·10-s − 4·11-s − 2·12-s + 2·13-s + 2·14-s − 4·15-s − 16-s + 6·17-s + 18-s − 2·19-s + 2·20-s + 4·21-s − 4·22-s + 2·23-s − 6·24-s − 25-s + 2·26-s − 4·27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.577·12-s + 0.554·13-s + 0.534·14-s − 1.03·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.447·20-s + 0.872·21-s − 0.852·22-s + 0.417·23-s − 1.22·24-s − 1/5·25-s + 0.392·26-s − 0.769·27-s − 0.377·28-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

Degree: \(2\)
Conductor: \(89\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{89} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 89,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.422304829\)
\(L(\frac12)\) \(\approx\) \(1.422304829\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.88636123831930, −18.53565555771721, −17.10273236683832, −15.57990391823508, −14.98281939160488, −14.14370325415045, −13.33146107000394, −12.30830966278088, −11.09736500054914, −9.547554941503177, −8.231468801983024, −7.853411931998762, −5.546797764618825, −4.176782576508255, −3.008392209118613, 3.008392209118613, 4.176782576508255, 5.546797764618825, 7.853411931998762, 8.231468801983024, 9.547554941503177, 11.09736500054914, 12.30830966278088, 13.33146107000394, 14.14370325415045, 14.98281939160488, 15.57990391823508, 17.10273236683832, 18.53565555771721, 18.88636123831930

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