Properties

Degree $2$
Conductor $90009$
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 − 4-s + 4·5-s − 4·7-s + 3·8-s − 4·10-s − 4·11-s + 4·14-s − 16-s − 2·17-s − 4·19-s − 4·20-s + 4·22-s − 4·23-s + 11·25-s + 4·28-s + 4·29-s − 2·31-s − 5·32-s + 2·34-s − 16·35-s + 6·37-s + 4·38-s + 12·40-s − 6·41-s − 2·43-s + 4·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.78·5-s − 1.51·7-s + 1.06·8-s − 1.26·10-s − 1.20·11-s + 1.06·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.894·20-s + 0.852·22-s − 0.834·23-s + 11/5·25-s + 0.755·28-s + 0.742·29-s − 0.359·31-s − 0.883·32-s + 0.342·34-s − 2.70·35-s + 0.986·37-s + 0.648·38-s + 1.89·40-s − 0.937·41-s − 0.304·43-s + 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90009\)    =    \(3^{2} \cdot 73 \cdot 137\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{90009} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 90009,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
73 \( 1 - T \)
137 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.76904680886731, −13.53126199896836, −13.27642743469532, −12.75759848188730, −12.50150973186903, −11.57175337288791, −10.68178587026152, −10.42690500461036, −10.02283704343169, −9.773263793844540, −9.155490500977104, −8.815181963921849, −8.259817137779170, −7.636267568003594, −6.879207268518205, −6.471310426517437, −5.960994476662598, −5.504258804580803, −4.846474271584550, −4.328408471959779, −3.434148937890831, −2.804864580621198, −2.209103738138233, −1.712865390126480, −0.6935849740620142, 0, 0.6935849740620142, 1.712865390126480, 2.209103738138233, 2.804864580621198, 3.434148937890831, 4.328408471959779, 4.846474271584550, 5.504258804580803, 5.960994476662598, 6.471310426517437, 6.879207268518205, 7.636267568003594, 8.259817137779170, 8.815181963921849, 9.155490500977104, 9.773263793844540, 10.02283704343169, 10.42690500461036, 10.68178587026152, 11.57175337288791, 12.50150973186903, 12.75759848188730, 13.27642743469532, 13.53126199896836, 13.76904680886731

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