Properties

Degree $2$
Conductor $51714$
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 + 4·7-s + 8-s + 6·11-s + 4·14-s + 16-s + 17-s + 4·19-s + 6·22-s − 5·25-s + 4·28-s + 4·31-s + 32-s + 34-s + 4·37-s + 4·38-s + 6·41-s + 8·43-s + 6·44-s + 9·49-s − 5·50-s + 6·53-s + 4·56-s − 4·61-s + 4·62-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 1.80·11-s + 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s + 1.27·22-s − 25-s + 0.755·28-s + 0.718·31-s + 0.176·32-s + 0.171·34-s + 0.657·37-s + 0.648·38-s + 0.937·41-s + 1.21·43-s + 0.904·44-s + 9/7·49-s − 0.707·50-s + 0.824·53-s + 0.534·56-s − 0.512·61-s + 0.508·62-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51714\)    =    \(2 \cdot 3^{2} \cdot 13^{2} \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{51714} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 51714,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.164165847\)
\(L(\frac12)\) \(\approx\) \(7.164165847\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.38683524824508, −13.92977469383552, −13.81615958767369, −12.99789783606899, −12.15779391196609, −11.99365458106343, −11.55083159740838, −11.05721054820635, −10.61262977672523, −9.656500119335563, −9.429787550699291, −8.668466024680720, −8.101041661697443, −7.569083117090960, −7.112698560734044, −6.391902802941246, −5.791355941370739, −5.412819194784533, −4.472598939392835, −4.310844597627686, −3.684307143648342, −2.858762875302694, −2.094661375296494, −1.377172860502349, −0.9461148615051682, 0.9461148615051682, 1.377172860502349, 2.094661375296494, 2.858762875302694, 3.684307143648342, 4.310844597627686, 4.472598939392835, 5.412819194784533, 5.791355941370739, 6.391902802941246, 7.112698560734044, 7.569083117090960, 8.101041661697443, 8.668466024680720, 9.429787550699291, 9.656500119335563, 10.61262977672523, 11.05721054820635, 11.55083159740838, 11.99365458106343, 12.15779391196609, 12.99789783606899, 13.81615958767369, 13.92977469383552, 14.38683524824508

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