Properties

Degree $2$
Conductor $8470$
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 − 5-s + 7-s − 8-s − 3·9-s + 10-s + 6·13-s − 14-s + 16-s − 2·17-s + 3·18-s − 20-s + 25-s − 6·26-s + 28-s − 6·29-s + 8·31-s − 32-s + 2·34-s − 35-s − 3·36-s − 10·37-s + 40-s − 2·41-s − 4·43-s + 3·45-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.223·20-s + 1/5·25-s − 1.17·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 0.169·35-s − 1/2·36-s − 1.64·37-s + 0.158·40-s − 0.312·41-s − 0.609·43-s + 0.447·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8470\)    =    \(2 \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{8470} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8470,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−17.34865696330956, −16.63280249078332, −16.05500888183225, −15.52735576701462, −15.05529262381296, −14.31888608298882, −13.65410840774739, −13.24195922976209, −12.22876614240727, −11.68307511294191, −11.27621041470977, −10.67469165206980, −10.18970420292551, −9.046331639181176, −8.780454827817629, −8.267889620477558, −7.616794734923788, −6.820567016120650, −6.128285639967756, −5.565486179673146, −4.618752397394013, −3.695169463770508, −3.091817928846595, −2.060782184560023, −1.140348346406679, 0, 1.140348346406679, 2.060782184560023, 3.091817928846595, 3.695169463770508, 4.618752397394013, 5.565486179673146, 6.128285639967756, 6.820567016120650, 7.616794734923788, 8.267889620477558, 8.780454827817629, 9.046331639181176, 10.18970420292551, 10.67469165206980, 11.27621041470977, 11.68307511294191, 12.22876614240727, 13.24195922976209, 13.65410840774739, 14.31888608298882, 15.05529262381296, 15.52735576701462, 16.05500888183225, 16.63280249078332, 17.34865696330956

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