Properties

Degree $2$
Conductor $20286$
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 − 2·5-s − 8-s + 2·10-s + 4·11-s + 2·13-s + 16-s − 6·17-s − 4·19-s − 2·20-s − 4·22-s + 23-s − 25-s − 2·26-s + 2·29-s + 8·31-s − 32-s + 6·34-s + 6·37-s + 4·38-s + 2·40-s − 6·41-s − 4·43-s + 4·44-s − 46-s − 8·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s − 0.852·22-s + 0.208·23-s − 1/5·25-s − 0.392·26-s + 0.371·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 0.986·37-s + 0.648·38-s + 0.316·40-s − 0.937·41-s − 0.609·43-s + 0.603·44-s − 0.147·46-s − 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20286\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{20286} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 20286,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9938466653\)
\(L(\frac12)\) \(\approx\) \(0.9938466653\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
23 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.77245481659425, −15.13125768866911, −14.84062963064195, −14.05436195422269, −13.43303797029648, −12.86135663783682, −12.18754879116998, −11.51515857672043, −11.37464362108695, −10.78502274901768, −9.977367356761333, −9.509901199640079, −8.768400984446375, −8.323855053523218, −8.031716875002008, −6.993516217621378, −6.550730773347249, −6.297699226655369, −5.136735618886370, −4.297504298515561, −3.966229542656161, −3.109382175570790, −2.234188005551476, −1.414474847258652, −0.4796550777390409, 0.4796550777390409, 1.414474847258652, 2.234188005551476, 3.109382175570790, 3.966229542656161, 4.297504298515561, 5.136735618886370, 6.297699226655369, 6.550730773347249, 6.993516217621378, 8.031716875002008, 8.323855053523218, 8.768400984446375, 9.509901199640079, 9.977367356761333, 10.78502274901768, 11.37464362108695, 11.51515857672043, 12.18754879116998, 12.86135663783682, 13.43303797029648, 14.05436195422269, 14.84062963064195, 15.13125768866911, 15.77245481659425

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