Properties

Label 2-2200-1.1-c3-0-95
Degree $2$
Conductor $2200$
Sign $-1$
Analytic cond. $129.804$
Root an. cond. $11.3931$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·7-s − 27·9-s − 11·11-s + 70·13-s + 10·17-s − 12·19-s + 84·23-s + 30·29-s − 72·31-s − 310·37-s + 18·41-s + 388·43-s + 516·47-s − 87·49-s + 298·53-s + 204·59-s − 210·61-s + 432·63-s + 432·67-s − 440·71-s − 46·73-s + 176·77-s − 616·79-s + 729·81-s − 740·83-s − 6·89-s − 1.12e3·91-s + ⋯
L(s)  = 1  − 0.863·7-s − 9-s − 0.301·11-s + 1.49·13-s + 0.142·17-s − 0.144·19-s + 0.761·23-s + 0.192·29-s − 0.417·31-s − 1.37·37-s + 0.0685·41-s + 1.37·43-s + 1.60·47-s − 0.253·49-s + 0.772·53-s + 0.450·59-s − 0.440·61-s + 0.863·63-s + 0.787·67-s − 0.735·71-s − 0.0737·73-s + 0.260·77-s − 0.877·79-s + 81-s − 0.978·83-s − 0.00714·89-s − 1.29·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(129.804\)
Root analytic conductor: \(11.3931\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2200,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
11 \( 1 + p T \)
good3 \( 1 + p^{3} T^{2} \)
7 \( 1 + 16 T + p^{3} T^{2} \)
13 \( 1 - 70 T + p^{3} T^{2} \)
17 \( 1 - 10 T + p^{3} T^{2} \)
19 \( 1 + 12 T + p^{3} T^{2} \)
23 \( 1 - 84 T + p^{3} T^{2} \)
29 \( 1 - 30 T + p^{3} T^{2} \)
31 \( 1 + 72 T + p^{3} T^{2} \)
37 \( 1 + 310 T + p^{3} T^{2} \)
41 \( 1 - 18 T + p^{3} T^{2} \)
43 \( 1 - 388 T + p^{3} T^{2} \)
47 \( 1 - 516 T + p^{3} T^{2} \)
53 \( 1 - 298 T + p^{3} T^{2} \)
59 \( 1 - 204 T + p^{3} T^{2} \)
61 \( 1 + 210 T + p^{3} T^{2} \)
67 \( 1 - 432 T + p^{3} T^{2} \)
71 \( 1 + 440 T + p^{3} T^{2} \)
73 \( 1 + 46 T + p^{3} T^{2} \)
79 \( 1 + 616 T + p^{3} T^{2} \)
83 \( 1 + 740 T + p^{3} T^{2} \)
89 \( 1 + 6 T + p^{3} T^{2} \)
97 \( 1 + 490 T + p^{3} T^{2} \)
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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

−8.626774150860195707272908019013, −7.51190524530924304978091960943, −6.67880055871588204224199725442, −5.90421545544309811136781554345, −5.37344578587189307145362735960, −4.06153373867016503505423095019, −3.30776209245288460664107103907, −2.51904405119199771990298319310, −1.12106020782597490521051181605, 0, 1.12106020782597490521051181605, 2.51904405119199771990298319310, 3.30776209245288460664107103907, 4.06153373867016503505423095019, 5.37344578587189307145362735960, 5.90421545544309811136781554345, 6.67880055871588204224199725442, 7.51190524530924304978091960943, 8.626774150860195707272908019013

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