Properties

Degree $2$
Conductor $2790$
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 + 4-s − 5-s − 8-s + 10-s + 4·11-s + 6·13-s + 16-s − 2·17-s + 4·19-s − 20-s − 4·22-s + 8·23-s + 25-s − 6·26-s − 6·29-s − 31-s − 32-s + 2·34-s − 2·37-s − 4·38-s + 40-s − 10·41-s − 4·43-s + 4·44-s − 8·46-s − 7·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 1.66·13-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s − 1.17·26-s − 1.11·29-s − 0.179·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s − 0.648·38-s + 0.158·40-s − 1.56·41-s − 0.609·43-s + 0.603·44-s − 1.17·46-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2790\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 31\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{2790} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2790,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.436678693\)
\(L(\frac12)\) \(\approx\) \(1.436678693\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
31 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−18.59456542714969, −18.35471766121626, −17.46674985651062, −16.88816491829030, −16.32266051193905, −15.70867537232271, −15.04563709037218, −14.50227315738617, −13.44889511464868, −13.13035846438895, −11.98005575233413, −11.47297246588434, −11.05495041056418, −10.22919184090326, −9.282486436639322, −8.835422004383419, −8.275089086557226, −7.203377114774192, −6.772212081828261, −5.920238820425020, −4.947659712277124, −3.737552092548992, −3.297224004906870, −1.755219263724784, −0.8859002356133358, 0.8859002356133358, 1.755219263724784, 3.297224004906870, 3.737552092548992, 4.947659712277124, 5.920238820425020, 6.772212081828261, 7.203377114774192, 8.275089086557226, 8.835422004383419, 9.282486436639322, 10.22919184090326, 11.05495041056418, 11.47297246588434, 11.98005575233413, 13.13035846438895, 13.44889511464868, 14.50227315738617, 15.04563709037218, 15.70867537232271, 16.32266051193905, 16.88816491829030, 17.46674985651062, 18.35471766121626, 18.59456542714969

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