Properties

Degree $2$
Conductor $2800$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 3·9-s − 4·11-s + 6·13-s − 2·17-s + 6·29-s − 8·31-s + 10·37-s + 2·41-s + 4·43-s + 8·47-s + 49-s + 2·53-s + 8·59-s − 14·61-s + 3·63-s − 12·67-s + 16·71-s − 2·73-s + 4·77-s + 8·79-s + 9·81-s + 8·83-s + 10·89-s − 6·91-s − 2·97-s + 12·99-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1.11·29-s − 1.43·31-s + 1.64·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s + 1.04·59-s − 1.79·61-s + 0.377·63-s − 1.46·67-s + 1.89·71-s − 0.234·73-s + 0.455·77-s + 0.900·79-s + 81-s + 0.878·83-s + 1.05·89-s − 0.628·91-s − 0.203·97-s + 1.20·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{2800} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.401898985\)
\(L(\frac12)\) \(\approx\) \(1.401898985\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
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

−18.57253169038225, −18.16437397656273, −17.68358075861944, −16.68170845057346, −16.26951127645794, −15.62511391693428, −15.06500081183901, −14.16547912680398, −13.57940150794464, −13.07936770236951, −12.37675015798141, −11.46630999545775, −10.85162413401503, −10.48952017451224, −9.335388238387631, −8.804167587109300, −8.107288137075358, −7.412972465405348, −6.241903473891820, −5.915783810064358, −4.995166840891652, −3.942483550249555, −3.075782092306130, −2.266399988755182, −0.7142897041111533, 0.7142897041111533, 2.266399988755182, 3.075782092306130, 3.942483550249555, 4.995166840891652, 5.915783810064358, 6.241903473891820, 7.412972465405348, 8.107288137075358, 8.804167587109300, 9.335388238387631, 10.48952017451224, 10.85162413401503, 11.46630999545775, 12.37675015798141, 13.07936770236951, 13.57940150794464, 14.16547912680398, 15.06500081183901, 15.62511391693428, 16.26951127645794, 16.68170845057346, 17.68358075861944, 18.16437397656273, 18.57253169038225

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