Properties

Degree $2$
Conductor $8400$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s + 4·11-s − 6·13-s − 2·17-s + 4·19-s + 21-s + 8·23-s − 27-s − 2·29-s − 4·33-s + 10·37-s + 6·39-s − 6·41-s − 4·43-s + 49-s + 2·51-s − 6·53-s − 4·57-s − 4·59-s + 6·61-s − 63-s + 4·67-s − 8·69-s − 8·71-s − 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s + 1.66·23-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 1.64·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.768·61-s − 0.125·63-s + 0.488·67-s − 0.963·69-s − 0.949·71-s − 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8400} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.410499848\)
\(L(\frac12)\) \(\approx\) \(1.410499848\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−16.84986575787150, −16.62152244087703, −15.75677001334407, −15.11627421083614, −14.64832588656122, −14.11078789045934, −13.17411506013176, −12.86070425739763, −12.08121066485812, −11.60096638853671, −11.18640176851329, −10.27335710822208, −9.645619440117420, −9.315801493448345, −8.551754264586085, −7.466936198453628, −7.122544799261485, −6.488268002074203, −5.749905616775195, −4.902317007104651, −4.502600055778668, −3.452144102556958, −2.726603190905346, −1.631161360266291, −0.6130493640677995, 0.6130493640677995, 1.631161360266291, 2.726603190905346, 3.452144102556958, 4.502600055778668, 4.902317007104651, 5.749905616775195, 6.488268002074203, 7.122544799261485, 7.466936198453628, 8.551754264586085, 9.315801493448345, 9.645619440117420, 10.27335710822208, 11.18640176851329, 11.60096638853671, 12.08121066485812, 12.86070425739763, 13.17411506013176, 14.11078789045934, 14.64832588656122, 15.11627421083614, 15.75677001334407, 16.62152244087703, 16.84986575787150

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