Properties

Label 2-2800-1.1-c1-0-39
Degree $2$
Conductor $2800$
Sign $-1$
Analytic cond. $22.3581$
Root an. cond. $4.72843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s − 2·9-s − 3·11-s + 2·13-s + 3·17-s + 7·19-s + 21-s + 5·27-s − 6·29-s + 4·31-s + 3·33-s + 8·37-s − 2·39-s − 9·41-s − 8·43-s + 6·47-s + 49-s − 3·51-s − 12·53-s − 7·57-s − 12·59-s − 10·61-s + 2·63-s + 7·67-s − 6·71-s + 5·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.554·13-s + 0.727·17-s + 1.60·19-s + 0.218·21-s + 0.962·27-s − 1.11·29-s + 0.718·31-s + 0.522·33-s + 1.31·37-s − 0.320·39-s − 1.40·41-s − 1.21·43-s + 0.875·47-s + 1/7·49-s − 0.420·51-s − 1.64·53-s − 0.927·57-s − 1.56·59-s − 1.28·61-s + 0.251·63-s + 0.855·67-s − 0.712·71-s + 0.585·73-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: $-1$
Analytic conductor: \(22.3581\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
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

−8.261780224373543856297235747442, −7.76136885452014769391786338919, −6.84082843689233258487775944714, −5.92654096551589702125781476334, −5.48776820834083428423049797351, −4.68405477142379205905809833243, −3.37835476194466045277789097411, −2.85425374545583595819763274865, −1.33133249048010073515633278194, 0, 1.33133249048010073515633278194, 2.85425374545583595819763274865, 3.37835476194466045277789097411, 4.68405477142379205905809833243, 5.48776820834083428423049797351, 5.92654096551589702125781476334, 6.84082843689233258487775944714, 7.76136885452014769391786338919, 8.261780224373543856297235747442

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