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Properties

Label 2-2800-1.1-c1-0-44
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  − 7-s − 3·9-s + 4·11-s − 2·13-s + 6·17-s − 8·19-s + 6·29-s − 8·31-s + 2·37-s + 2·41-s − 4·43-s − 8·47-s + 49-s − 6·53-s − 6·61-s + 3·63-s − 4·67-s + 8·71-s − 10·73-s − 4·77-s − 16·79-s + 9·81-s + 8·83-s − 6·89-s + 2·91-s + 6·97-s − 12·99-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s − 1.83·19-s + 1.11·29-s − 1.43·31-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.768·61-s + 0.377·63-s − 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.455·77-s − 1.80·79-s + 81-s + 0.878·83-s − 0.635·89-s + 0.209·91-s + 0.609·97-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: $-1$
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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
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 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−8.476850680857876089528395940037, −7.76192246751474515867810853341, −6.73925079130933177784315595590, −6.20937036396666016095385773228, −5.41709512685248088965000889554, −4.42263366581703790425444246040, −3.54031487268152506468532472221, −2.71865927886904183893896021093, −1.51739479053058957038284358713, 0, 1.51739479053058957038284358713, 2.71865927886904183893896021093, 3.54031487268152506468532472221, 4.42263366581703790425444246040, 5.41709512685248088965000889554, 6.20937036396666016095385773228, 6.73925079130933177784315595590, 7.76192246751474515867810853341, 8.476850680857876089528395940037

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